More than anything else, I would like more save slots.

Besides that;
Having specialized actions for deep dungeon would be a nice touch, like how the PvP actions work.

It would be nice to have access to the pvp Adrenaline limit break instead of the normal 4-person party one.

Having more unique encounters would be fun; encountering random enemies with mechanics, or perhaps boss-type creatures being randomly generated would be cool, especially if they had special drops or rewards.

Special room types would be really fun as well- maybe a room full of toxic puddles like in Aurum Vale; a room covered in slippery ice; a room full of poison mist that damages you while you're in it; a large maze room, maybe the size of four rooms, with extra treasure chests inside; a room where AoE indicators will appear and shortly after a stalactite will fall on that location, dealing damage, just neat variations.

I would also love to be able to summon my Chocobo companion in the dungeon.