There are lots of times when a set admits more structure than you might have originally noticed. A pretty elementary example is when a combinatorial or set-theoretical device turns out to admit an action by a group $G$, making it into a $G$-set. This is what underlines the unreasonable effectiveness of group theory in combinatorics. So a laymen's example might be something as simple as the 15-puzzle or the Rubik's cube. Or maybe you run into a group that itself admits a "hidden" action from some other group, further illuminating its structure. Or maybe your abelian group turns out to be an $R$-module for some ring $R$ of interest. All of these phenomena have occurred many times in the development of mathematics.
On a (only slightly) different front, maybe you find that the set of points on a geometric curve admits a shockingly important group structure. Ditto for other geometric objects like the space of line bundles on a projective variety. Or maybe a topological space (or a group) admits the structure of a Lie group. Or maybe a space parameterizing some interesting collection of objects turns out to actually be a fine moduli space for such objects.
The list goes on and on. In fact, I think it would be a reasonable stance (among many other valid such stances) that the search for hidden structures is the process of doing mathematics.