ct.category theory - Does this concept have a name, and what are some of its applications?

One reasonable notion of “smallest” here could be “initial”?

So for an object $W$ of $mathbf{D}$, we would look at the iso-comma category $(F downarrow_{cong} W)$, where an object is an object $X in mathbf{C}$ together with an isomorphism $i_X colon F(X) cong W$, and a map $f: (X,i_X) to (Y,i_Y)$ is a map $f : X to Y$ with $i_Y cdot F(f) = i_X$. Then we could define a “minimal $mathbf{C}$-object generating $W$” as an initial object of $(F downarrow_{cong} W)$. (Or possibly weakly initial?)

Idempotent functors of the kind you're describing often fall into one of two classes: reflections or co-reflections. Reflections are more usual: they're functors that come from an adjuntion $mathbf{C} leftrightarrow mathbf{D}$ where the left adjoint goes from $mathbf{C}$ to $mathbf{D}$, the free way to $mathbf{D}$-ify a $mathbf{C}$-object. The abelianisation of a group, the field of fractions of a ring, the Stone-Cech compactification of a space, sheafification of a presheaf, are all examples of this. The unit of the adjunction gives a natural map $X rightarrow F(X)$.

I can't think of many examples of reflections where $(F downarrow_{cong} W)$ will have an initial object for all (or even most) $W in mathbf{D}$. In the “field of fractions” case, for instance, $mathbb{Z}$ is an initial object in $(mathit{Frac} downarrow_cong mathbb{Q})$, but fields of fractions $k(x)$ will have no minimal generating ring. (High-falutin' explanation: $(F downarrow_{cong} W)$ has all connected colimits that $mathbf{C}$ does, but not other limits/colimits in general.)

What about when $F$ (let's call it $G$ now) is a co-reflection, i.e. comes from a right adjoint to the inclusion of $mathbf{D}$? This situation typically looks a little different: the “group core” of invertible elements in a monoid, for instance; in this case the natural map goes the other way, $G(X) to X$.

In this case, there'll always be an initial element of $(G downarrow_{cong} W)$: just $W$ itself! (This is dual to how in the reflection case, W is always a “maximal generating object” for itself, i.e. a terminal object of $(F downarrow_{cong} W)$.) On the other hand, here perhaps a terminal object is a better sense of “minimal” (“maximally quotiented”, or something): taking the sheaf of germs of a bundle is a co-reflection where this might be an interestingly non-trivial question?

Some examples, of course, aren't quite either reflections or co-reflection, like “algebraic closure”, which is nearly a reflection but not quite, because of the automorphisms. This case will look, I suspect, similar to the reflection case…