I always thought it'd be a cool idea to take this sort of thing to extremes. Base a whole 3d/first person PC game on it...

The box doesn't give away what the game is about. You open it and install, and are presented with something innocuous (like this phone.) Let the user figure it out. The new Myst game was something like that, but it sort of relied on you playing the first few to flesh out the back story. I'd rather have the details filled in by "found clues". Also, the puzzles shouldn't be jarringly illogical to the situation. Rather they should be natural mysteries. Figuring out how a strange and alien works machine, rather than aligning colored pebbles on an (illogicially placed) gameboard to open a door.

Basically a wild adventure game with no spoilers other than word of mouth.