My favourite weakness that I ever gave a character was that I did a supervillain character who would act against the interests of the party at least once per game (or help them out but in a way that they found morally reprehensible). But before he tortured or killed anyone, he was compelled to give a monologue explaining exactly how he was going to kill them, which meant that they had a chance to escape.

Which is probably not useful for your setting, but I like weaknesses in terms of "they can do this cool thing but here is a catch."