I agree with some of the comments that "profinite set" is not a standard term. But you can certainly look at the category of pro-(finite sets). In other words, begin with the category $Set_f$ of finite sets and functions. Then one can form a category $Pro(Set_f)$ as the projective completion of the category $Set_f$; it is the full subcategory of the category of functors from $Set_f$ to $Set$, consisting of objects isomorphic to projective limits of systems of finite sets. In other words, the objects of $Pro(Set_f)$ are (not necessarily representable) functors from $Set_f$ to $Set$, which are inductive (viewing finite sets via Yoneda as functors from $Set_f$ to $Set$ switches arrow directions) limits of representable functors from $Set_f$ to $Set$. I'm sure one should be careful about some smallness/universe issues to make this precise.

The category $Pro(Set_f)$ is equivalent to the category of compact totally disconnected topological spaces. This elaborates on Leonid's answer.

The reference for this somewhat highbrow answer is a paper of Gaitsgory and Kazhdan, in GAFA, titled "Representations of algebraic groups over a 2-dimensional local field".