I would love to hear other people's character's One Unique Thing. I've been fooling around with some characters while I wait for a chance to run the game. I've only made a couple, but so far I have...
A Wood Elf Bard who's One Unique Thing is that He hatched out of an egg laid by The Green.
Which is cool, but I think the Relationships with Icons helps explain it a lot better.
He Has a positive relationship with the Three and the Priestess, who both believe that he will fulfill a prophecy that involves the Green, but in different ways. He has a conflicted relationship with the Elf Queen, who feels that I should live, but isn't sure that it won't cost her her life. Due to being a bard. I also have a negative relationship with the Crusader, who sees me as an evil spawn that should be taken care of... when there's time.
The other character I've made is a Human Fighter. His one unique thing is While trying to woo the Diabolist to save his village, the Diabolist stole his heart and put it in a glass bottle.
His relationships is a Conflicted one with the Diabolist (Hey, she has his heart. Literally), and a Positive one with the Prince of Shadows.
i'll add more when I make more characters, and I have no idea if a DM would let me use those things, but I find it a fun exercise.