What should I do if one of the screws for a TV mount breaks?
Luckily your mount has slots for the bolts so you have a bit of room to work.  Get another (stronger, higher quality) lag bolt and put it in right beside the broken one.  

Leave a tiny bit of room between them if you can, but make sure it still solidly hits the stud.

Predrill the hole.  To find the right size, hold the bolt up behind your drill bit.  The drill bit should cover the entire shaft, but not the threads.  Make sure you drill to the same depth the bolt will go to.

Rub the bolt threads on a candle or bar of soap to lube it up a little if you think it is going to be hard to drive in again.

Drive the bolt in slowly.  Don't use a drill unless it has a low torque screwdriver setting.  You might want to tighten it with hand tools anyways so you don't risk snapping this one too.  There probably isn't room for a third try!