I am building a new exhaust system. It won't be fancy, but I would prefer anvil like durable.

How thick should the exhaust headpipe flange be? I am thinking 3/8 is probably overkill, would 5/16ths be the best choice?

A little overkill won't hurt. 5/16 should be plenty but 3/8 is only another 1/16. Not enough difference to worry about. I'd get the most available. Now if you said 1/4 or 3/8 is definitely go 3/8

The exhaust flange I am replacing is thicker than 1/4 but it bends and warps easily.

3/8 it will be, as the local supply house does not have any hot rolled flat in 5/16...