What do you look for in a stock pot?

Thick bottom and large are your two major needs.

A good stock pot can be useful well beyond stock. Think "Dutch oven" and the utility factor multiplies considerably. In that light, I'd look for the same attributes you'd want in any pot or pan (mine are All-Clad tri-ply and I routinely brown on the stovetop and transfer to the oven for braising). For stock, you don't need anything fancy whatsoever; restaurant supply stores are an excellent source for inexpensive aluminum pots for the purpose.

It needs to be durable, definitely NOT non stick, and large enough to hold all the stuff you're going to put in. I always laugh when I see sets with a 5 or 6-quart "stock pot." No way you can make stock in a pot that size. Altho it's fine for heating up the finished soup.