Is there a theory of simplicial cohomology with coefficients in a non-abelian group ? I've found next to nothing on Google so far...
I'm interested in particular in the cohomology of graphs with coefficients in the symmetric group $S_3$...
The idea of nonabelian cohomology in dimensions higher than 1 goes back to Dedecker, and you can search his papers on MathSciNet. But even for dimension 1 there are variant ways of expressing it. These are discussed in the paper Fibrations of Groupoids; the point is that 1-dimensional cohomology involves derivations $d: G \to A$ but these are conveniently seen as sections (which are morphisms!) of the projection $G \rtimes A \to G$. (I learned this idea from Philip Higgins.)
Dedecker's insight for dimension 2 is that $H^2(X,A)$ for $A$ nonabelian is not really functorial in $A$ since the definition involves automorphisms of $A$. So he generalises it to coefficients in a crossed module $\alpha: A \to P$.
The further development of this idea is if that your coefficients are going to involve crossed modules then it is helpful to define a construction $\Pi$ from groups $G$ or simplicial sets $X$ to crossed complexes, which involve crossed modules, so that a cocycle with coefficents in a crossed module $\mathcal M$ is essentially a morphism $\Pi X \to \mathcal M$; and the cohomology set is essentially just homotopy classes of morphisms. This puts lots of seeming complications into the construction of the functor $\Pi$ and allows one to use the techniques of model categories for homotopy theories.
The full details of this are available in a book published in 2011 by the EMS and with a pdf available here.
One interesting reason why all this development can be done is by working directly with filtered spaces rather than just spaces. But many useful spaces, and certainly simplicial sets, come equipped with a filtration, so this is no great hardship, except to make the conceptual leap of looking more at the structure of the space.
As suggested by Aaron, you may need only the case of $H^1$. But note that graphs have lots of vertices so you may really need groupoids rather than groups, and I have a paper called Groupoids as coefficients which I suspect has hardly been cited!