What changes in the sheaf theory of topological spaces with the "étale topology"? The customary site structure on the category of topological spaces has covering families given by open covers. What "happens" if we refine this topology and let any jointly surjective family of local homeomorphisms be covering? Refining the Zariski topology has lots of interesting consequences, so I wonder what kind of thing happens for topological spaces.

Absolutely nothing changes. The category of etale sheaves on a topological space $X$ is equivalent to the category of ordinary sheaves on $X$. The reason is that ordinary open subsets (i.e., maps $U\to X$ which are homeomorphisms to some open subset of $X$) are cofinal in all etale maps to $X$, since you given any etale map $p:Y\to X$ you can cover $Y$ by open subset on which $p$ is a homeomorphism to its image. So, any etale sheaf is determined by the values it takes just on open subsets of $X$, and those values just need to form an ordinary sheaf.

The point of using etale covers in algebraic geometry is that the Zariski topology is too coarse, and an etale cover of schemes does _not_ locally look like just a cover by open sets but instead is more general even locally. But for topological spaces, there is no difference because etale covers are just defined using the open sets of the topology.