What could be a cool name for the DRK relic?

[Firepower]

From previous FF games (ignoring Apocalypse, Ragnarok and Liberator)

There is Masamune, although technically a Katana you have a great sword design that Auron used in FFX and rikku as DRK in FFX-2. (in FF3 DRK and NIN both used darkblades of which the Masamune was one)
If you argue that SAM (if added in future) should have it, there are still quite a few alternatives for SAM like the Amanomurakumo or Kogarasumaru.

More likely, since there is already similarities to Cecil's DRK from FF4 and it being a weapon given to DRK after completing the additional part of the starter quest for the job in FFXI, it would be the Deathbringer.

[Aurelinaus]

Tyrfing, Caladbolg, Gram, and Valkyrja because I think it just sounds cool.

[Kuroyasha]

Call it Viagra

[Lustrate]

I would go with umbra

[krazyrs]

Go classic with deathbringer

[BlaiseArath]

Its likely that they'll either get a new quest for relic weapons (Ala FFXI and "Relics" / "Mythics") and won't be on Zodiac weapons but will be on the new ones... or they'll create Zodiac Weapons for them. If they do create Zodiac Weapons for them, its likely the name will come from a weapon within Final Fantasy Tactics universe.

With that in mind, Valhalla is the likely name, as it is the "Legendary Fell Sword" from Final Fantasy Tactics. Arondight, Balmung, and Chaos/Deathbringer are the other Fell blades of that, but I do believe the final version will be Valhalla.

[OpticOcelot]

I really like the name Umbra for the relic

[Amnesia]

The Batblade

[Alacran]

Dainsleif

Dáinsleif ("Dáinn's legacy") is king Högni's sword, according to Snorri Sturluson's account of the battle known as the Hjaðningavíg.
When Heðinn offers him compensation for the abduction of his daughter, Högni replies:

'Thou hast made this offer over-late, if thou wouldst make peace: for now I have drawn Dáinsleif, which the dwarves made, and which must cause a man's death every time it is bared, nor ever fails in its stroke; moreover, the wound heals not if one be scratched with it.'

Another translation: Dain's Heirloom A sword made by the dwarf dain 1, according to snorri sturluson. This sword was cursed Once drawn, it must kill a man before it could be returned to the sheath. A blow from this magical sword never failed to kill or cause a wound that never healed. To rescue his daughter Hild, the legendary warrior and king Hogni drew Dainsleif to kill her abductor, Hedin Hjarrandason.

[Zophar]

Soul Edge /nod