I like the idea of Cena being turned heel via stopping Undertaker's streak, but there's only one problem with it, as was previously mentioned. It's a big problem, big enough to stop this from happening. As was said, Cena has beaten all of the top guys in the WWE. The Streak is the one thing he hasn't touched. By giving him that one thing, after beating all of the top talent, you're more or less giving him the distinction of the best WWE wrestler of all time.
Think about that - John Cena - best EVER in the WWE. Does that not sit well at ALL with anyone else? For the amount of time he has been in the WWE, you can't give him the streak. You either give it to someone who has paid their dues and is at the end of their career (a la HBK, HHH, etc.) or give it to someone who has not been in the WWE terribly long. You also have to give it to someone who's going to be a heel, which again narrows down the field. If you're going with someone newer, you also have to be certain that they're staying with the company and not bolting part way through their career (see: Lesnar).
There are two (young) people currently on the WWE roster that I would give the streak to. Not DiBiase. Not McIntyre or Sheamus (they haven't proven themselves enough, though I'd give the edge to Sheamus). You give it to either The Miz (loyal to the WWE, great heel, would boost him to the stars) or CM Punk (also great heel, a bit longer in the company, would solidify him as being one of the top heels in WWE).
In order to give it to the Miz, he'd have to improve his inring skills. In order to give it to Punk, you'd have to make sure that his Straight Edge gimmick can last, as that would be the main storyline against the streak.
But I digress, I've strayed to far from the topic. Don't give it to Cena!