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…