After a point, thickness comes into play because the pinhole becomes more of a tunnel. The longer the tunnel, the more fall-off at the extremes, and if really thick, you will get full vignetting. I had used 0.002" brass stock for a few that worked well, and then succeeded in making a couple out of 0.001". This past April, I discovered one of those later ones had a crack in the foil! I theorize the sanding, plus using some sort of nasty chemical to blacken the finished plate, may have weakened it a bit too much.
My first pinhole plate (at least in the 21st century) was made with soda can stock. I would say my later results have been better, but I also blackened that first one by smearing a felt permanent marker around it, and it could be that left a small blob or two in an unfortunate location. A careful dimple-and-sand job actually creates a knife edge for the hole, so I would expect the sensitivity to material thickness to be less important as long as you don't start using 0.032" or some such thing.
My 1.3 (after tax) cents,