Is the sheaf cohomology only a sheaf for flabby sheaves?

Question

Let $X$ be a topological space and let $\mathcal{F}$ be an abelian sheaf on $X$. Choose an arbitary open cover $\{U_i\}_i$ of $X$. Then the Mayer-Vietoris long exact sequence is $$0\rightarrow \mathcal{F}(X)\rightarrow \prod_i \mathcal{F}(U_i) \rightarrow \prod_{i,j}\mathcal{F}(U_i\cap U_j)\rightarrow H^1(X,\mathcal{F})\rightarrow \prod _iH^1(U_i,\mathcal{F}\vert_{U_i}) \rightarrow \prod_{i,j} H^1(U_i\cap U_j, \mathcal{F}\vert_{U_i\cap U_j})\rightarrow \ldots$$ Then a sufficent and necessary condition for $U\rightarrow H^1(U,\mathcal{F}\vert_U)$ to be a sheaf is that the map $$\prod_i \mathcal{F}(U_i) \rightarrow \prod_{i,j}\mathcal{F}(U_i\cap U_j)$$ is surjective. My question is: When is this true? A sufficent condition for this would be that $\mathcal{F}$ is flabby since $H^1(X,\mathcal{F})=0$. However, this is not very interesting as sheaf. So I'm wondering if there is a ``good/interesting" case when $H^1(-,\mathcal{F})$ is a non-trivial sheaf? More generally, one can ask the same question for sheaves on an arbitrary site by replacing a topological space with a site and $\mathcal{F}$ by a sheaf on this site, and the Mayer-Vietoris sequence exists because of the Cech to derived spectral sequence. Are there good conditions to put on this site or sheaf to make $H^1(\mathcal{F})$ a non-trival sheaf on this site?