One thing missing from the current dungeons are various ways to activate different hidden paths or puzzles in general. Thinking back to past FF titles and even XI, there were pressure plate puzzles, different levers you had to pull to visit certain floors within mines or even open a door on the other side of the dungeon.
It would be great if the current dungeons (especially Mistbeard Cove since it seems like it will be a pirate base eventually, Tam-Tara deepcroft, since it looks like some old forgotten ruins, and the different mines in Thanalan) as well as the near future dungeons made use of various puzzles or plates/levers to change things up while traveling within them. Perhaps add some extra lower areas to some current dungeons in order to make use of these mechanics (would be easy to tie some company excavation mini-story-lines to these).
Random treasure chests within the dungeons that may contain rare items, potions, ethers, would also aid in bringing the Final Fantasy feel as well as giving people reason to explore every inch of a dungeon if possible.