That would actually be kind of cool. I am not much into modern adventure games (though back in the Lucas Arts days I was a big fan), but I would play this one just because I like the world of Bioshock so much.
However, I think you could have a Bioshock prequel with a lot of action. Through audio diaries we learn of all sorts of crazy stuff going on in Rapture. CIA were sending spies, smuggling was running rampant, citizens were attacking small girls in order to get a symbiotic seas slug out of their stomachs, freaky mutant in diving suits were made in order to guard the little girls. Then of course there was the civil war. Yeah, the multiplayer used this period for its setting, but an actual game set at that time could be cool.
I really loved the first Bioshock, and unfortunately have been too busy to beat the second one, but really enjoyed what I have played of it. But I am weary of it getting too popular and destroying its self. The release of the second one made me feel confident that they could keep a good thing going, but I am still cautious.
I actually think putting you in the role of a CIA agent during the early days of Rapture could be pretty cool. You could be sent to find out as much about the place as possible. For enemies you could have local Rapture law enforcement, Fontain's mob, and even Subject Delta could make an appearance as a boss enemy.
Something I never had time for, but was still very intrigued by was the Something in the Sea web game. It made Rapture sound just as interesting from the outside looking in, as from the inside. I'd almost like to see a game take that angle. Or stretch it even further and take things beyond Rapture. Maybe some citizens of Rapture left right when things started to get ugly and began their own city, full of many of the same innovations as Rapture on some remote island someplace. Your character could shipwreck on this island and begin trying to figure out the mysteries of the strange technology he find everywhere while trying to survive against hoards of madmen. Or maybe the Splicers on the surface found ways not to go insane or oversplice and remain normal, but are still opposed to outsiders and intend to eliminate you.
I'd actually like to see something like that above all else. Audio diaries could explain how they foresaw the fall of Rapture and took their families and left. As pretty as Rapture is, we have seen it, and I'd love to see what they could do in a different setting. Fighting a Big Daddy in the jungle, with trees burning from the garnade he just lobbed is something that would get me excited. Seeing a Big Daddy walking along a sunny beach hand in hand with a Little Sister is something that would get me as excited as when I played Bioshock for the first time. Maybe the guys on this island figured out another way to get ADAM without using small girls as hosts to the sea slugs. Honestly, with a world as interesting as that of Bioshock, the possibilities are endless. Unfortunately, imagination is often lost when a franchise outgrows itself. I hope that is not the case with BS.
As much as I like BS 2, I am not sure I would want to play a third game unless they do something really different.