The thread I refer to here about character-centric games also concerns a PC with amnesia, where the player asked the GM to come up with her backstory. A player saying "can you give me a mysterious past?" can be a chore, but it can also be one of the best darn plot hooks you'll get. If it’s good enough for Luke Skywalker, River Tam and Jason Bourne...
Of course, it's up to you (with or without some input from the player) to make this interesting.
Back when White Dwarf was about RPGs, issue 75 had an article about mystery in games, including PCs being mysteries to each other and even themselves: "The classic example is the character with no parent(s) - who are they? This poser is not very interesting if player characters aren't special - he may be the heir to... a baker's shop!"
The trick is knowing how far to go. This can vary with the game, the genre, the character and the player.
An amnesiac character in Vampire suggests dark secrets, possibly removed from her mind deliberately, an absentee sire, as well as the potential shock of meeting those who knew her as a mortal... or indeed learning she has not been a mortal in a lot longer than she thought. Doctor Who has its own variety of possibilities. A high-fantasy amnesiac could also discover that her memory loss was mystical, but is less likely to be appalled by what she learns. The heir to a baker's shop would suit Warhammer but not Star Wars, but not everybody who says "can you give me a mysterious past?" wants something as big as "no... I am your father."
One option would be to sketch out some possibilities, perhaps in deliberately vague terms, for the player to consider.
Of course, all this makes the revelations a surprise to the player as well as the character. This makes for more natural reactions, but there's a lot to be said for the player being in on the joke. They'll often make things far worse for their character to be more entertaining if they know what the "worse" would be.