In algebraic set theory a la Joyal and Moerdijk, the subset relation *is* taken as fundamental, with membership only being a derived notion (specifically, the cumulative hierarchy is taken to be the free partial order with small joins and an abstract "singleton" operator. The order corresponds to subsethood, and x is defined to be an element of y just in case the singleton operator applied to x is a subset of y). I can never quite grasp what it is that mereology is supposed to be all about as a contrast to set theory, but if it's just a matter of viewing subsethood as more elementary a concept than membership, well, there you go.