Honestly? Nothing. NOTHING.
Seriously, BioShock was a masterpiece that should be preserved and kept alive through memories. Up to five sequels, as far as I hear, and there is NO way that they can be good. After such an amazing and well-made game as BioShock, I really don't think that the companies should be shitting all over it.
After all, Rapture is pretty much done for, and there are two endings, both of which could have had a dramatically different effect, and therefore they would have to choose one or the other to actually make a sequel work.
And a prequel? Half the fun of the game was piecing together the story yourself by finding diaries and such. BioShock was a very psychological and generally amazing game, and I really want to see it stay that way, rather than turn out as another Resident Evil/Silent Hill.
So, in conclusion, no sequel, no prequel, leave it how it is, bitches!