I love all the FF series I've played, especially the little things that make it very interesting.
Let's start with my favorite: FF8.
Aside of the numerous requests about Triple Triad (which I support wholeheartedly) how about a long chain of Levequests with a rating?
Like the SeeD mission, when doing this long levequest which can consist of numerous tasks (escort, kill enemies, use items..) You will be assessed on how well you perform. (Killing wanted targets, timeliness, not running away, special conditions! Like killing X-ATM-092) and then rated from 0-100 depending on the criteria you fulfilled and then get experience points and treasure chests or tomestones and gil accordingly.
FF9, I believe someone had posted an idea about festival of the hunt, which I love tremendously. Just probably make it level sync'd so everyone can have equal chances of getting top rank, which is usually a decent card or accessory..
FF7, I do hope all or most of the gold saucer games make it in our gold saucer. The boxing game, the motorcycle game.. Chocobo breeding and racing, all the arcade games, even the touch of the haunted inn.. The laser shooting! Oh god, that was glorious.
FF4, well.. not much of a minigame-y aspect to it, but I loved how there were hidden chests on hidden paths. We have one in the Sunken Temple of Qarn, but I'd love to see more of these..
FF10 I liked the monster collection kinda thing. I for one would love a massive hunting log that we have right now, for example, for a more interesting leveling experience make us kill like 30 kinds of monsters, and 10-20 of each kind. And the monsters you killed gets into some sort of bestiary and then we can view images and some background story to them..
And one FF that I'm sure not many have played or even heard of, but in my opinion is the king of minigames in the gaming world is the Final Fantasy Fables: Chocobo's Tales. All the minigames there were awesome. I'd kill for all of these to be incorporated to XIV, but with its own touch of lore.