My nominee is the definition of a topology. I don't know the history of the subject but my impression is that this was the result of a collective effort. The definition with collections of subsets closed under finite intersections and arbitrary unions is not the kind of thing one would get up one morning and decide to consider. It seems to me this was an example of category where the definition of morphisms was clear (continuous maps) but where finding the right notion of object was not obvious.

Perhaps a sign of a good definition is that it leads to other good definitions. Those could arise in order to remedy some flaws of the original one, like Grothendieck's topos. They could also arise as particular cases of the original definition like Schwartz's definition of the topology on $\mathcal{D}$ underlying the notion of distribution mentioned in Alexandre's answer.