My vote for show to redo, this one. While I agree with Mass Effect being a good television show, Bioshock on the other hand would not make a good TV series, without a lot of changes. In fact I do not believe that it should have even qualified for this debate based on your number 2 qualification you set in the beginning. It does not really have a strong base of characters that demand a show. There are really only 3 strong characters in the game, but they only work as very important side characters. While those characters might be great they are really not enough for a TV series. I cant think of one TV show where there is only a single main character. So either they would have to make up some new characters or change one of those three to become a main character.
Don't get me wrong, I think the world of Bioshock has the potential to make a great TV series. It is just that TV series would have to be quite different than the game character wise.