Categories $J$ such that limits of shape $J$ commute with filtered colimits in sets are called L-finite. There are several known characterization of them: see the nLab page about it. The page refers to Robert Paré, Simply connected limits (pdf). In particular, Proposition 7 says:
A category is L-finite iff it has an initial subcategory which is finitely generated.
There's always some ambiguity in the term "finite limit" because you can consider the source of a "diagram" to be either a graph or a category. This matters not for the notion of limit because of the free/forgetful adjunction between graphs and categories, but the free category on a finite graph may not be finite; it is however L-finite, as you'd expect. This is a situation where it seems more natural to view diagrams as graph morphisms rather than functors.
Jardine has recent a paper called "The Verdier hypercovering theorem", based on an earlier paper called "Cocycle categories", which you should look at if you haven't.
He doesn't exactly say it this way, but it looks like the following is true: given presheaf such that $X$ is locally fibrant, then you can define the "hypercover cocycle category" $H_{hyp}(U,X)$ to be the category whose objects are diagrams $U\xleftarrow{p} V\rightarrow X$, where $p$ is a hypercover (i.e., local trivial fibration). Then the nerve of $H_{hyp}(1,X)$ is a simplicial set with the homotopy type of $\Gamma(\hat{X})$, and more generally $NH_{hyp}(U,X)\approx \Gamma(U,\hat{X})$.
This is in the spirit of what you are asking for. Perhaps it can be modified to involve simplical mapping spaces, if that's what you really need.
Added:
In any model category $M$, for objects $X$ and $Y$, Jardine defines the "cocycle category" $H(X,Y)$ to be the category whose objects are spans $X\xleftarrow{f} A\rightarrow Y$, where $f$ is a weak equivalence, and whose maps are $A\to A'$ compatible with the projections to $X$ and $Y$.
If the model category $M$ has functorial factorizations, is right proper, and has the property that weak equivalences are preserved under all finite products, then you can show that if $Y\to Y'$ is a weak equivalence, then the evident map $H(X,Y)\to H(X,Y')$ is a weak equivalence on nerves; in fact, it is a simplicial homotopy equivalence. This is basically Lemma 3 of Jardine's Cocycle categories paper; he describes it as a proof that $\pi_0H(X,Y)\to \pi_0H(X,Y')$, but in fact his proof gives (assuming factorizations are functorial) an explicit map $H(X,Y')\to H(X,Y)$ and simplicial homotopies from the composites to identity, making it a simplicial homotopy equivalence.
(If $M$ is simplicial presheaves on a site, and if $Y$ is locally fibrant then the evident inclusion $H_{hyp}(X,Y)\to H(X,Y)$ is also a simplicial homotopy equivalence, also by using a factorization argument; this is exactly Lemma 5 of Jardine's Verdier hypercovering paper.)
To understand the homotopy type of $H(X,Y)$, it suffices by the above remarks to consider the case that $Y$ is fibrant (in the model category structure). That $H(X,Y)$ computes the homotopy type of the derived mapping space follows from results in the papers of Dwyer and Kan on hammock localizations (Function complexes in homotopical algebra and, especially, Calculating simplicial localizations; Prop. 6.2 of the latter is what you want.).
Best Answer
I think there are actually three possible things that you might be asking, but the answer to all of them is no. Suppose that G is a strong generator in a cocomplete category C. Then you can ask:
Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$? (If so, then G is called dense in C.)
Is every object some colimit of a diagram all of whose vertices are G? (If so, then G is called colimit-dense in C.)
Is C the smallest subcategory of itself containing G and closed under colimits? (If so, then G is a colimit-generator of C.)
The category of compact Hausdorff spaces is a counterexample to the first two. It is monadic over Set (the monad is the ultrafilter monad, aka Stone-Cech compactification of the discrete topology), and hence cocomplete, and the one-point space is a strong generator. But any colimit of a diagram consisting entirely of 1-point spaces must be in the image of the free functor from Set (the one-point space being its own Stone-Cech compactification), and hence (since that functor is a left adjoint and preserves colimits) must be the free object on some set. However, not every compact Hausdorff space is the Stone-Cech compactification of a discrete set.
As Todd pointed out in the comments, though, CptHaus is the colimit-closure of the one-point space, since every object is a coequalizer of maps between free ones (because the category is monadic over Set); thus it isn't a counterexample to the third question. Counterexamples to the third question are actually much harder to come by, and in fact if you assume additionally that C has finite limits and is "extremally well-copowered", then it is true that any strong generator is a colimit-generator. I think there's a proof of this somewhere in Kelly's book "Basic concepts of enriched category theory," and a brief version can be found here. However, without these assumptions, one can cook up ugly and contrived counterexamples, such as example 4.3 in the paper "Total categories and solid functors" by Borger and Tholen.