As far as story goes, I'd have to go with FF6, though 4, 5, 9, and 10 all rank up there. I loved the cast of characters, and I'll admit I have a bias towards a larger cast - which is why FF6 tends to win over 4/9/10 and especially 5. It's by far the only game that I would love to see a sequel to, just to see the characters again. The story itself wasn't overly weird or esoteric either, though Kefka might have been. I guess I liked the simplicity of it - the protagonists fight a corrupt empire, and eventually a megalomaniac leader who comes into ultimate power. While I liked 9 and 10, I'd really like to see Square go back to something a little simpler as far as the overall arc goes.
For gameplay, I must admit I'm fond of 10, though 4 also ranks high up there. The reason for this is fairly simple. In an RPG, a character is defined through two means. The first is story, and story is good, yes. However, in an RPG, a good majority of the time is spent in battle, and thus a certain amount of characterization can and must occur in battle as well. The issues I have with 5/7/8 is that the characters don't have any personality in battle. Frex, it didn't matter if I used Barrett or Aeris or Cloud or Yuffie or whomever, because all I needed to do is switch over the Materia and I had the same exact character in battle, with very few, minor differences. However, in 4 characters had a role in battle, and it was the same in 10. 6 had much the same, and I supposed 1 did as well, though really the first one was all battle and little story.
As far as "systems" go (and it seems all the rage to have systems nowadays) FF5's job system holds a place in my heart. And any game with a world map rocks my socks... I hate the recent trend in RPGs to do away with it. I much prefer a system akin to Wild Arms, where there's incentive to actually search the map. Puzzle Boxes own my soul.
Finally, as far as dungeons go, I have a deep loathing of every FF dungeon. It's the one thing that has remained constant that really needs to be changed. Again I'm going to compare to Wild Arms, which has pretty boring dungeon structure overall, but it mixes puzzles in with the tedium of random battles. Which as an aside, really need to go. I'm very glad to not see them in FF12, though I'd prefer a Chrono Trigger style approach. Another great example would be Lufia 2. The combination of puzzles and trying to avoid enemies really breaks up the tedium of endless walking. My only complaint, and I haven't seen any console RPG show this, is that even with those, dungeons as a whole just seem to be boring. The overall design just falls flat, and I'd like to see a little innovation as far as the basic layout goes.
[EDIT] Also, mini-games. Someone really needs to tell Square that endless repetition of tedious button-pushing doesn't make a mini-game. It makes at best an annoyance, and at worst an exercise in extreme frustration. Contrast that with the card game in FF7, which was simple and easy. I'd point towards the Suikoden series for some interesting ideas.