Character wise, from the games I've played (spoilers, obviously):
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IV: Cecil and Kain. Cecil's wrestling with orders and being loyal to a country only pushing for power (with a phony king to boot), and his subsequent fight for redemption and acceptance by those he wronged (Palom, Porom, the town of Mysidia, Rydia). Kain's constant fight for control of his own mind and will form Golbez, and again the redemption following his freedom. I like IV's story because to me, it was the last one to take risks with the story. V did it with Galuf, but IV wasn't afraid to make you watch party members go out like heroes. Granted not all of them were dead for good, but you don't know this till nearly the end of the game. Even Edward, who's useless as shit every time you take him along (unless you play the re-release on DS or PS1, then he's a little less useless) proves he's something special.
VI: Cyan and Setzer. Cyan's tragedy of a life from the moment Kefka camps the army by Doma to his journey for revenge and accepting the loss of his loved ones and home. It's a beautiful, albeit short, story, but it cemented him as one of my favorite FF characters (even if he's a huge pervert in the Japanese version). Setzer seems like a suave gambler, but one scene was all he needed to really stand out. The descent to Daryl's airship in the World of Ruin, and the scenes played as you see his life with someone he admired like a hero. Everyone's stories were pretty great in this game, even Strago's (an old hero who wants to prove it one last time).
VII: This one is kinda hard as I feel like this is where the character development started to decline compared to the old days. I'd have to go with Cid here. Shinra takes everything from him, but through it all he refuses to give up the sky he loves, no matter how surly he's become over the years. Tifa is another good one, doing everything for a man who's not even the one she remembers from childhood. Zack is a close third, after playing Crisis Core he turns out to be the baddest of dudes in the FFVII universe.
Tactics: The whole goddamn thing. I love this story, the characters, everything. It's just so good.
VIII: It's all pretty much junk to me here, I hate this game. Don't think I really had a favorite.
IX: It's been a loooooooong time, so I honestly can't remember here.
X: As much as people hate him, and as annoying as he is early on, Tidus. He starts out a whiny, annoying bitch, but eventually realizes there's a lot for him in this world, even if his is just a dream. The ending was pretty solid, and I really wish FFX-2 had been a new FF with new characters, because I felt bringing him back cheapened the original X's ending.
XII: I've actually not completely finished this, but Basch and Baltheir have quickly become my favorites.
XIII: My 360 died awhile back so I never finished this, but really nobody stood out for me much.