I'm not sure how big a deal the WaPo story about Marco Rubio "embellishing" the facts of his family history will turn out to be for Rubio's career; that his parents left Cuba before Castro instead of fleeing Communist oppression might matter to some, but it's not as if he faked military service or gave himself bogus educational credentials.
The oddity of the Rubio situation is that I don't recall such an obvious VP frontrunner in any previous cycle. Now, preseason Veepstakes is notoriously silly; after all, guessing the pick even when there's just a few weeks to go and we know who is doing the picking rarely works out well. And the usual caveat applies: the bottom of the ticket doesn't really matter very much in November. So I'm not speculating about whether Rubio will actually get the nod. But it is, I think, worth pointing out that near as I can tell there's been a pretty solid consensus that Rubio is the obvious selection, and that such a consensus is unusual. My guess is that this story doesn't really shake the current consensus -- although whether everyone's expectations now have anything to do with who actually gets the pick is unknown and unknowable.
What does matter about VP selections is that it's a major boost towards actually becoming president someday; a lot of vice presidents have become president one way or another, and even losing VP nominees are often serious presidential contenders in future cycles.
The other thing to say about VP selections is that they've changed over time as nomination politics changed. It used to be that the second spot was a bargaining chip that a nominee could trade for support at the convention. That meant that it was far more of a party selection than it has become. Now, it's purely the pick of the nominee, made well after the nomination is wrapped up. There's still some party constraint; the nominee certainly doesn't want important party factions to get upset, and at the extreme case it's possible that the convention delegates could cause a very visible fuss, although remember that delegates are usually selected for their loyalty to the nominee. And of course the nominees themselves are creatures of very partisan candidacies.
At any rate, as I said I suspect that the Rubio expectations will survive this with very little damage; as Dave Weigel argues, there's not really a lot to this particular story. Whatever that eventually means to the nominee.