Etymology of terms.

[Rongway]

Geirskogul

Just a small note on this one, because I often see people mistakenly abbreviate it "GK": it should be parsed geir skogul, not geirs kogul.
As for its pronounciation, r+s in Norse-based languages become "sh", even when separated by a space, so something close to "gish-kugl" is a far better approximation for it than "geer-sko-gool". If you're interested in hearing a native speaker of Icelandic and Norwegian speak it, this YouTube user says it while reciting part of the Völuspá, about 12 seconds in.

[Cilia]

Echidna is also known as the mother of monsters in a lot of references I've seen.

Also, theory on how Ferdiad died. His hide may not be penetrated, but he could still be beaten to death.

Well, yes... Echidna was the mate of Typhon, a beast so terrible Zeus had to personally step in and defeat him. The two spawned many of Greek myth's horrible monsters.

As for Ferdiad... it's not that it's unknown, just kind of... err... takes context and is a bit gruesome.

In mythology, the Gae Bolg is super deadly not because of its quality as a weapon, but because it could kill from a single scratch. After piercing the opponent, the Gae Bolg would spread no less than thirty barbs through its victim, killing them instantly. (Could also only be thrown with the toes, but I digress.)

Since Ferdiad's hide was impenetrable, Cu Chulainn instead gave his foster brother an unwanted rectoscopy - he shoved the Gae Bolg up Ferdiad's ass, triggering its enchantment and killing him.

Let's continue, ehh...

PRIMALS
Ifrit: high-ranking, dangerous djinni from Arabian myth.
Titan: progenitors of the gods from Greek myth.
Garuda: avian creature (bird, birdlike creature, or humanoid bird) that serves as Vishnu's mount.
Leviathan: a sea monster from the Bible. Identified as the patron demon of the sin of Envy in some works.
Ramuh: unknown, possibly derived from Rama, hero of the Ramayana and incarnation of Vishnu.
Shiva: creator and destroyer deity of Hinduism.
Ravana: ten-headed demon king of Hindu myth. Could not be killed except by mortal hands, which is why Vishnu incarnated into Rama.
Bismarck: named after Otto von Bismarck, a Prussian statesman who was instrumental in unifying Germany, or the World War II era ship which bore his name and was sunk after only one offensive operation.
Lakshmi: Vishnu's wife in Hindu myth; goddess of wealth.
Susano (Susano'o, Susano-o): Japanese god of storms. Full name "Takehaya Susano'o no Mikoto." One of three gods born along with his sister Amaterasu and brother Tsukuyomi when Izanagi cleansed himself after journeying into Yomi, the underworld.

Manipulator: a robotics term for a device used to move things without direct contact.
Brute Justice and components: a big shout-out to Transformers, specifically the Decepticon combiner Bruticus. Each component of Brute Justice uses a slightly modified name of one of Bruticus' components and assembles exactly the same way (Onslaught is the head and torso of Bruticus, Onslaughter is the head and torso of Brute Justice, etc); Brute Justice is stated to "lack autonomy" by his Triple Triad card, much like Bruticus is very dumb.
Cruise Chaser: designation Blassty. A reference to an early and obscure Squaresoft game, Cruise Chaser Blassty, which was also the inspiration for IX's Ark summon, hence why the two share many moves.
Arrhidaeus: half-brother of Alexander the Great who went with him on his conquests. Became a figurehead king after Alexander's death.
Alexander: Macedonian king who conquered a massive empire, which largely fell apart after his death. Literally means "protector of man," which likely contributed to his depiction as the "Holy" summon in the franchise and status as the sole unambiguously benevolent primal in-story.

Sephirot: emanations of the Kabbalah.
Sophia: lit. "wisdom" in Greek.
Zurvan: creator god of an extinct branch of Zoroastrianism, who came before Angra Mainyu and Ahura Mazda.
Bahamut: gigantic fish that supports the Earth in Arabian myth.
Odin: a major deity of the Norse pantheon.

CRYSTAL TOWER
Thanatos: Greek god of death. Generally peaceful despite contemporary depictions.
(King) Behemoth: another gigantic monster from Christianity.
Acheron / Phlegethon: rivers that flowed into Hades, the Underworld, in Greek myth.
Scylla: a sea monster from Greek myth, situated opposite her counterpart Charybdis on a narrow naval channel, making approach extremely hazardous.
Glasya Labolas: one of the 72 demons from the Lesser Key of Solomon.
Amon: patron deity of Thebes, and a major god in Egyptian myth. Alternatively, another of the 72 demons from the Lesser Key.
Angra Mainyu: essentially the Zoroastrian equivalent to the Devil.
Cerberus: Hades' guard dog from Greek myth.

CELESTIAL DRAGONS
Midgardsormr: "Midgard Serpent." One of Loki's three children from Norse myth. A serpent so massive it is capable of encircling the Earth (Midgard) and biting its own tail; archenemy of Thor.
Hraesvelgr: "Corpse Swallower." A giant from Norse myth that took the form of an eagle, who caused the wind to blow when he beat his wings.
Nidhogg: "Malice Striker." A dragon from Norse myth who lives beneath Yggdrasill, the world tree, and gnaws at its roots.
Ratatoskr: "Drill Tooth." A squirrel from Norse myth that carries messages between Nidhogg and an eagle perched atop Yggdrasill.
Bahamut: see Primals.
Tiamat: a primordial sea goddess from Mesopotamian myth.

For a more general appearance...

Atomos: Greek for "undivided." Likely a reference to atoms, the building block of real-world matter.

Aether: the "fifth element" in some ancient alchemical beliefs. Believed to be the air the gods breathed, associated with the heavens and emptiness.

Einhander: German for "one-hander." A reference to a PS1 era Squaresoft game of the same name, this boss uses a gimmick similar to that game's.

[WhiteArchmage]

Just a few corrections/expansions to Cilia’s overview (and some of the others, and some from myself):

Arjuna (not Arjun, I realize I’m being pedantic) holds the Gandiva, which is indestructible and has 108 strings (108 being a “holy” number in Buddhism and Hinduism).

Longinus is mostly referred to as the name of the roman soldier that pierced Christ, which then became confused with the name of his spear (Tactics Ogre: The Knight of Lodis plays with this with the Longicolnis, supposedly meaning “long horn” …it doesn’t mean that), it is also known as the Spear of Destiny.

Necronomicon means “Book of the Death” so of course “Draconomicon” means “Book of the Dragons.”

Ragnarok is a little more confusing as it can mean (from old norse) Doom of the Gods, the most generally accepted translation IS “Twilight of the Gods”, helped by Wagner’s Gotterdamerung (literally "Twilight of the Gods" in German).

An Ifrit is specifically a Fire djinn, the arabs believed the four elements were represented by different djinn (or genies), the Ifrits holding fire, the Marids water and the Djinni Wind, Humans were the earth genie, who unfortunately don’t have shape-shifting powers like the other 3.

Ramuh could be Rama, or he could be Lahmu from Akkadian myth (beardad man with a staff and and a red sash); or it could be an amalgam of Rama and Vishnu.

Sophia, besides meaning wisdom in greek is also the Gnostic goddess of, well, wisdom.

While Bahamut does indeed come from Babylonian myth, his appearance as a giant Dragon can be chalked up to D&D, where he appears as the Silver God of Good Dragons, FFI rolled with it and a dragon he stayed.

Sephi’s attacks are also emanations of the sephirotic tree (Yesod, Da’at and Tif’eret) while Ein Sof is the name for The Infinite (God, so to speak).

Durandal was Roland’s sword, with which he attempted to break so it wouldn't be taken from him after his death…and he ended up cutting a boulder in half instead.

Joyeuse HASN’T appeared, but has been mentioned as one of the first Sultansworn’s weapons. Joyeuse was Charlemagne’s blade, which, if I’m not mistaken, was said to change colors “30 times a day”.

Not sure if someone’s mentioned it but Benediction also comes from Latin, pretty much means “Good saying” or “Speak Well” but its actual meaning is “Blessing”.

[Singularity]

Indeed it does, and an addendum to that, the Bandersnatch enemy found in the Chocobo Forest in the Dravanian Forelands is also from the same source (it was also a weird dog-like enemy in FFIX) - the 'bandersnatch' was a monster mentioned in Jabberwocky as a mysterious yet ferocious and dangerous beast - indeed, two of the FFXIV enemy's attacks, 'Fulminous Jaws' (a fire-like attack), and 'Catching Claws' (a swipe attack), are named after lines about the Bandersnatch in Jabberwocky: specifically, "with fulminous jaws" and "claws that catch". When I first saw that I smiled wryly immediately picking up on the reference.

Slight correction to this, the word is "frumious" and is another of Carroll's nonsense words. In the words of the author himself (via Wikipedia) - "[T]ake the two words 'fuming' and 'furious'. Make up your mind that you will say both words, but leave it unsettled which you will say first. Now open your mouth and speak. If your thoughts incline ever so little towards 'fuming', you will say 'fuming-furious'; if they turn, by even a hair's breadth, towards 'furious', you will say 'furious-fuming'; but if you have the rarest of gifts, a perfectly balanced mind, you will say 'frumious'."

Frumious also the only term used to describe the bandersnatch in the poem. "The jaws that bite" and "the claws that catch" are both attributed to the Jabberwock itself:

"Beware the Jabberwock, my son!
The jaws that bite, the claws that catch!
Beware the Jubjub bird, and shun
The frumious Bandersnatch!"

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Another random item: Tetragrammaton, the WHM lv.60 skill is a Greek word literally meaning "composed of four symbols" and is a name given to the name of God in ancient Hebrew, which is written without vowels (and thus is of uncertain pronunciation) YHWH. You may notice that the skill animation involves a circle of four orbs, presumably in reference to this.

[Enkidoh]

Ah yes, I really should have not relied upon my terrible memory for that and verified the attack names! Thanks for clarifying that Singularity.

Now for more FFXIV terms and concept origins! :3

Two FATE NMs in Heavensward are taken from Australian Aboriginal mythology - firstly the FATE 'Run, Run Away' in the Chocobo Forest in the Dravanian Forelands against a vingaroon NM named 'Yowie', and the FATE 'Waugyl, Waugyl, Waugyl' at the Monsterie in the Churning Mists against.. a puk NM named "Waugyl', both are creatures from Aboriginal mythology.

The Yowie is a mysterious humanoid monster (somewhat like Australia's version of Bigfoot or the Yeti) found in Aboriginal myths across Australia and the Waugyl is a serpent deity from the Aboriginal Noongar tribal group from the South West area of Western Australia. When I saw both these NMs in FFXIV I smiled and was pleased that SE are drawing from myths from all countries, including Australia.

The glyph for the god Nald'thal the Traders is a cowry shell. Cowry shells were actually used as currency in ancient times (due to their beauty and rarity), a cowry being a type of sea snail, and thus this makes a lot of sense for the mark of a deity associated with trade, commerce and yes, money and it's acquisition.

[Eloah]

I also would like to make an addendum to the info that Cilia has generously provided.

Titan: The second generation of gods in Greek mythology. The Titans themselves are gods, and are the pregenitors of the Olympian Gods most people are familiar with. The first generation being the Primordial Gods or as ironic as it might sound the Primal Gods, lol
Titan's connection to the element earth most likely stems from a modern interpretation of the origin of the name coming from the greek word τίτανος which mean white earth, clay, or gypsum. This interpretation stating that the Titans were "white clay men" or men covered in white clay for their rituals.

Eos: Titan Goddess of the dawn, sister of Helios and Selene
Selene: Titan Goddess of the moon, sister of Helios and Eos
Helios: Titan God of the Sun, brother of Eos and Selene
Hyperion: One of the first generation of Titans and the father of Eos, Helios, and Selene

And here is a nice tidbit, that might be more unintentional than intentional.
Aether: A Primordial God, the personification of the upper air, and in some stories the father of Uranus. Aether in ancient Greek means light, and the upper air he "governs" is the air only the gods breathe. So basically the Primals are sustained by Aether, sounds a bit familiar.

[Shadotterdan]

One more thing. Regarding Shiva in the FF series and why it is so different from the Hindu Shiva. Apparently the katakana used in FFIII were supposed to be read as "Shiver" but FFIV put it as Shiva and it's been kept ever since.

Also, more info on Glasya Labolas, referred to as the president (or earl) of hell, the author and captain of bloodshed and slaughter, tells all things past and future, knows all sciences, gains the love of all who know him, can incite homicide and make people invisible. Appears as a dog with griffin wings.

[Eloah]

One more thing. Regarding Shiva in the FF series and why it is so different from the Hindu Shiva. Apparently the katakana used in FFIII were supposed to be read as "Shiver" but FFIV put it as Shiva and it's been kept ever since.

That's actually inaccurate, the katakana for Shiva is シヴァ, which would only be read as Shiva. The whole Shiver thing came about because of peoples misunderstanding of the Japanese writing systems and Shiva's ice element. True the words would look similar シヴァ(Shiva) vs シヴァ-/シバ-(shiver). But there was no mistranslation, the name has always been Shiva.

[Cilia]

Many thanks to those who provided additional info. It's not that I didn't know or couldn't be bothered to look it up, but just wanted to keep things pretty concise since I tend to infodump. (Also Bahamut does owe his modern depiction as a dragon to D&D, but he is the platinum dragon god of the good metallic dragons. Likewise Tiamat owes her modern depiction as a dragon to D&D, where she is portrayed as a five-headed evil dragon goddess, with one head representing each variety of chromatic dragon - red, blue, green, black, and white.)

Enough showing off my nerdiness. A few more things:

Hecatoncheires: three giants from Greek mythology stated to have a hundred hands and fifty heads. Sealed in Tartarus until Zeus freed them to help with the Titanomachy, after which they became the guards of Tartarus. Among their number were Kottos and Gyges, the two hecatoncheir bosses in Copperbell Mines. The third named hecatoncheir, Ouranos (final boss of Copperbell HM), is another name for Uranus, Greek god of the sky and, as in-game, father of the hecatoncheires (but not one of them himself).

Vidofnir: a rooster said to sit atop Mimameior, a tree similar to Yggdrasill, in Norse myth.

Vedrfolnir: a hawk said to sit between the eyes of the nameless eagle perched atop Yggdrasill.

Vishap: a dragon from Armenian mythology, said to live on Mt. Ararat.

Rangda: demon queen of the Leyaks, floating heads with entrails hanging out behind them, in Balinese mythology. Opposed by Barong.

Gyascutus: a mythical beast with longer legs on one side for traversing hills, also known as "sidehill gougers." Also a genus of beetle.

Macuahuitl: a weapon traditionally from Mesoamerica before the European conquest of the continent. A wooden sword using embedded obsidian shards to create a blade keener than steel.

Irminsul: a sacred, pillar-like object in Saxon pagan religion. Its exact purpose remains unknown.

[Singularity]

The third hecatoncheir from the myth, Briareos (also called Aigaion) does also appear in the game. He's a FATE boss that spawns in the tunnel between Vesper Bay and Parata's Peace.