Terra, Celes, or Locke from FFVI. All of them have troubled pasts that they need to overcome. Terra has no memories and she's
, Celes is cold as ice and doesn't trust anyone due to the fact that she was raised as a weapon of the empire, and Locke has to overcome the death of a loved one that he blames himself for. This all helps the story in a wonderful way, and helps it to not only have the best story in a Final Fantasy game (maybe even in an RPG), but to be the most enjoyable FF game of all time.
Runner up would have to be Rydia from FFIV. Along with Palom and Porom, she's one of the few characters you give a shit about when she dies (well... more accurately, "dies", as in Gandalf), despite the fact that she wasn't extremely useful as a kid. Don't tell me that you cared about Edward or Yang more when they were scattered away from your party when Leviathan attacked. Or that you didn't squeal with glee when she came back aged 17 with 90 billion times her original usefulness. She had a hotness factor that was through the roof and a great personality, too!