I am extending my back patio and am planning on reinforcing the new slab. I live in South Louisiana. Should I use wire mesh or rebar for this?
Rebar is better in about every use case. The mesh will have to be installed directly in the middle of the slab. The pourers will have to do a lot of work to make sure that the mesh stays there while being poured and moved. I have seen so many mesh jobs where I can assure you that the mesh just sunk to the bottom and was more or less useless - well maybe 20% functionality.
If the contractor can be trusted to do the mesh right - they should be charging your extra for creating supports for it, then mesh vs rebar probably is of little consequence on a backyard patio. You are paying to not have future cracks - and really in Louisiana this should be easier due to the lack of frost cycles.
Make sure the patio is poured to at least 6" and go with rebar unless the contractor explicitly details their method of keeping the mesh on center of the pour.