Sort of an answer to my own question that I just stumbled across: A "stack over a category S" $S$" is a functor from some other category into S that satisfies various properties.
If these properties are similar or dual to the ones that a presheaf must satisfy in order to be a sheaf... then I would say that the notion I was talking about in my question was that of a "pre-stack," and that the answer is "Yes, they are useful... they give rise to the notion of stacks!" EDIT: Considering my original question simply asks if the notion of functors into a category being useful, it doesn't really matter if the property of a stack that separates it from any old functor into $S$ is related to the property of a sheaf (that separates it from any old functor out of $S$)... What matters, I think, is that stacks are a type of functor INTO $S$ as opposed to out of it, and they are, indeed, useful. But it would be nice if someone could clean this answer by erasing this sentence and replacing it with some "big-picture" description of what (if anything) a stack over $S$ tells us about $S$... in relation to, say, a sheaf on $S$. Am I still incomprehensible? It is rather late...
Of course, I may be spouting nonsense... so I'm making this answer a community wiki in the hopes that someone who knows something about stacks can verify or falsify what i've written, and hopefully expand.
(Meta-note: I put this as an answer because I wasn't really sure what to do... I don't think it properly belongs as an edit to the original question... it's basically a different question "are stacks basically what I was thinking of?" but it was sufficiently related and similar to my original question that I don't think it merits a whole NEW question... so I put it here as community wiki. If this is not the correct thing to do, then I trust a moderator or someone much wiser than me will take down this post!)