The damsel-in-distress thing has been such a staple of video games and movies for so long because the premise always works like gang busters. Heroic dudes need things to be heroic about and a chick that needs help always fits the bill. You think Luke Skywalker would've been swinging on ropes and shooting stormtrooper if he'd known Leia was his sister? Hate to say it, but probably not. Going to war over a girl you want to snog is a different kind of motivation than fighting for a blood relative that you've just met.
Anyway, Bioshock Infinite puts Elizabeth --not Betty, not Liz, not Beth-- as the motivation for Booker's adventures in Columbia. She's got powers, somebody wants her and Booker will get paid for getting here. Simple. Or at least it should be. Check out Bioshock Infinite's stylized action in this latest trailer and try to convince yourself this game won't be awesome. You can't do it.