My best guess will be John Flaherty
There are three potential choices on the roster in Ken Huckaby, John Flaherty, and Josh Bard.
Ken Huckaby is, quite likely, the best defensive catcher of the three. He has a very good SB Caught percentage, and there has to be a reason he has stayed around as long as he has. However, he has some problems. Namely, he is a horrific hitter. The best hit he ever had was when he hurt Derek Jeter (not in a good way, but contact wise). Seriously, if he gets 120-150 plate appearences, he will hurt this team severely at the plate.
Josh Bard is the second choice. He's far younger (28) then Huckaby (35) or Flaherty (38). He was received in trade from the Indians in the Crisp/Marte swap. Like Varitek, he's a switch-hitter, but he isn't nearly as competent in hitting as Flaherty. He's cheap, and I'm guessing he's not out of options, so they could send him to Pawtucket.
Flaherty really has nothing over Bard. He's older (well, that may not be bad for a backup), he isn't as good defensively or offensively, and the Yankees wouldn't even take him as a backup.
Why do I say Flaherty then? They signed him to a deal, either 650k or 850k, and I don't think they'll waste that in AAA if Flaherty competes with the other two.
This is all assuming they don't use Tek there. I don't think they will, he plate is already full enough, and he took a sharp decline in September even when he got every 5th game off. Making him play every game would simply hurt him too much.