It's clear that the étale topology is closed related to some form of inverse function theorem. Let me give some reasons. (Also, see the comments here.)
(1) A morphism $f:X\to S$ between smooth varieties over $\bar{k}$ is étale at $p\in X$ if and only if $\mathrm{d}f_p:T_pX\to T_{f(p)}S$ is an isomorphism.
(2) A smooth morphism $f:X\to S$ of relative dimension $n$ is, étale locally, an open immersion into $\mathbb{A}_S^n$. (This is basically the structure theorem of submersions.)
(3) If $f:X\to S$ is finite étale, then every $s\in S$ has an étale neighborhood $V\to S$ such that $X\times_S V\to V$ is a trivial cover.
(4) If $\bar{x}$ is a geometric point of a scheme $X$, the ring $\mathcal{O}_{X,\bar{x}}$ is strictly henselian. This is the ring of germs of functions étale locally at $\bar{x}$, and the Hensel lemma is very closed related to the inverse function theorem.
However, I still don't know what's precisely the inverse function theorem in this context. I would love if it were the following statement, which is clearly not true.
Let $f:X\to S$ be a morphism between smooth varieties over $\bar{k}$ such that $\mathrm{d}f_p$ is an isomorphism. Then there exists an étale neighborhood $V\to S$ of $f(p)$ such that $X\times_S V\to V$ is an isomorphism.
P.S.: I know that this question is perhaps too simple for MO, and that it's similar to this. I still think that here's the best place and that it's sufficiently different to the other question. Feel free to close the question, if you think otherwise.