A couple of my favorites:
Iron Man: Billionaire playboy weaponsmith ends up on the wrong side of his own creation and is forced to invent a new technology to save his own life. He has a change of heart and realizes that he can also use that technology to fight the forces of evil. I love redemption stories.
Beast: Born with physical gifts beyond the limits of normal human achievement, he uses his extensive knowledge of biochemistry to make himself even stronger, faster, and more agile. The fact that it turns him into a blue-furred monster hardly phases him at all.
Captain Britain: Choose one of the following artifacts - the Sword of Might, or the Amulet of Right. Captain Britain proved he was worth of the title with his choice.
And honestly, after the 90s, I'm pretty much a fan of any origin story other than "born a mutant".