What type of backer board do I need to use to attach a stone veneer around a fireplace?
Drywall is out of the question due to weight, plywood might not give you the proper adhesion. 

I would start with cement board, then attach the metal lath to that.  After the metal lath is secure, you'll have to apply a scratch coat of a sand and cement mix before applying the stone.

Here is a good guide for applying stone veneer.