Some relatively standalone personal thoughts (doesn't come crashing in and ruining pvp/pve balance or progression for example lol):
I think a chocobo hot and cold could be fun and relatively straight forward to add (compared to say "revamp housing" lol), and to easily reference it to a current active system in another MMO basically look at WoW's Archaeology - except FFXIV's would be 100x cuter.. cause Chocobos. Like in FFIX I would love to see it culminate into a Chocobo Paradise/Heaven visit lol. Combine this with the below chocobo raising and you might get your fourth gathering job although that isn't necessary for the idea (would be an interesting mix of different types of items - some could be based on that sort of Archaeology idea and others more natural gathering style like the other gatherers, and adding levels could allow for development to the hot cold game and a mini game in that maybe).
Glamour/item log another straight forward but highly useful one (although SE seems to have a lot of data issues). With the log you could encourage a big time recycle on your content, with a reward for doing it - and maybe lore bits on some items.
Bestiary log (classic RPG/FF thing), again SE might have memory issues, besides lore and EXP (cannibalize the other log that requires killing monsters into this one), could also be leading towards a coliseum / monster making / fighting system like FFX's arena (doesn't have to end at itself). May work with the monster master idea they had to turn Blue Mage into a solo non-dungeon job (please don't do that to Blue Mage, but go ahead and make the content lol).
Aesthetic growth- its come up in many forms like a tree, sphere grid, glyphs, unlock page, FFIX item learning, merit points, but a way to increase player customization through progress. I feel this would be good in general for FFXIV because this game's end game for a majority of players seems not to be raiding but just glamour/fashion. So being encouraged to learn how to change grit from red fractals to smokey shadows or so when you cast a spell as BLM you float instead might be fun (optional modifications to your character, like perhaps glowing eyes in combat, and probably be best to add some sort of limit so even if you have 100 things unlocked you can't use all 100 at once lol). Might be able to add other race's animations for somethings as an unlock, like unlocking the female dark knight idle pose as a male (or male for female), instead of a sword to the chest the casting animation for paladin is to just praise dat sun, etc, etc.
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Personal desires that are really big and greedy:
Getting everyone a house and making it more baseline might be able to open up some interesting things, looking at Wild Star (not only game that does this though, it's just a good example of it doing well) where you can basically design your own pocket world and many things you collect can make your play experience more powerful (exp/rested buffs, building mines, training areas, mini games, gardens, raid portals, and more). As a leader of an entire country pretty much the WoL shouldn't have issue owning their own plot of land. Current housing system is like owning a house in down town, the land is so small, overpriced (if you're buying from players), and limited. An important thing to add is if everyone gets a place to call their own that then they can start adding really cool / seen as more mandatory to complete features to housing because there is no issue about "not enough houses, not fair to do". Can leave the neighborhood experience as an optional externally for people and when they use the door they enter their pocket world, although personally couldn't care less about the neighborhood experience since even on the largest server it seems mostly dead except for like one house lol.
Like for example FC needing to build a foundry or forge building, leading to them adding an airport hub in their floating airbase so they can then work on a very large physical ship that can be taken on raiding missions into Garlean territory. Where you act as a sort of pirate (basically take the FFXI pirate attacks, the besieged, except you're the pirates). Would push more gil around if more people buy and sell as more people can actually take part, also more money to SE because of the cash shop (those housing items in the CS). Although of course making it work in the first place would probably cost SE a bit...
I'm not so sure about a whole Garrison system for each player (WoW) but I do think adding that sort of thing for an entire Free Company that they can work together to build would be a really good idea, only having the workshop as a place to project together is pretty bad imo. Also consider looking at Guild Wars 2's Guild Halls which are /massive/ areas with upgrade-able buildings too. Being able to upgrade a larger space with more impactful visuals, as well as doing reputation building for your FC in a meaningful way over a long period of time would be very nice (imo).
True chocobo raising, like baby chick to adult, less npc doing it/cut scene only, with customization including variations of feathers as seen in all other FFs (fluffy chocobos of FFXIII). Definitely allowing them to be used in squadron allowed dungeon content. Good idea to include other features though like retainer missions on them, just so they become overall useful - perhaps raising for certain NPCs/GC for gil/token/reputations to encourage the system not to be a one off and done thing (You stock the immortal flames with strong war steeds sort of thing). With the everyone gets a house with actual land thing I could even have my own chocobo ranch *squeeeeeeee* (FF7~)
My moogle buddy, or at least a fuller companion system and a moogle as an option for one, and make one of them more useful housecarl type thing. Be able to flute him like FFIX for a shop I can slowly upgrade through quests/reputation with other npcs, ask him to take items from me to put into my vault, and other useful stuff like teaching them how to make sure my plants dont die (perhaps puts them in stasis, upgrades increasing how long). If I'm near my vault we can work together so they pull the items out of the vault for me automatically and "hands" them to me, so I can craft from my inventory and the vault ("two" person crafting in select areas, perhaps sometimes giving small buffs as a cute here take this moment, reduces lost items, nothing so powerful that it changes crafting balance so you must use them or fail - just helpful randomness). I miss my FFXI moogle. . . >.<;.