A few things come to mind:
1. Keep a copy of the key and if it's not claimed after X days then take it back.
2. Don't have a checkbox to show taken as that just stops anyone from bothering. Even if they're disappointed it's better they explore how to get them as that increases knowledge.
3. Make sure they know they have to spend them before someone else and not use that address further. It'll be a counterproductive if they imported that key into a wallet and let it sit such that someday more of their own money ends up on that address and someone else with the key steals it.
Actually just creating this "sweepstakes" ticket is an exercise in trying to optimize explaining Bitcoin attractively such that someone would even bother.
The idea of a game is good...
How about you post 
sudoku puzzles that when solved have a 81 digit number that seeds a key?
That way only the ones who take it and solve it and plug it into bitaddress.org get the key. It raises the entry level bar but also is more of a challenge and so makes more sense to people as a winning.
You can generate bulk sudoku pdf or SVG puzzles with this google code project here,
http://code.google.com/p/sudo2svg/You could also put these online in blogs/craigs list etc. It's the Bitcoin Sudoku Challenge... can you figure it out and WIN?
I think this would be attractive to those who like puzzles and nerdy stuff like crosswords etc but who aren't yet involved in Bitcoin or even programming stuff.
SOLVE a Sudoku for cash - get free Bitcoin.