Well, the main character from Second Sight is unique in the fact that he has psychic powers, but has no idea why he has them (at first), and also doesn't have the typical "I can do whatever I wish to" mindset of most psychics.

Rydia from Final Fantasy IV is also unique because despite being...seven years old (I think), she had the maturity level of somebody triple her age, and didn't have the typical "is weak and is captured by the main villain" complex of some other game characters.