Of course you want it to be as thin as possible, but rigid at the same time.
Some rawhide's are somehow harder and make thinner fletchings, so i would prefer those. My friend tanned his own deer hide and ended up with superior fletchings.
I just measured my arrows and those had 0,5mm thick (1/50 of an inch) goat hide fletching. With good, stiff hide you might be able to lose as much as half of the thickness... i think.