There is the "flabbification" or "flasquification" functor used in the Godement resolution. Namely, given a sheaf $\mathcal{F}$ on $X$, let $\mathcal{F}_x$ be the stalk at a point $x$. Then we define the sheaf $\Phi(\mathcal{F})$ to have sections $$\Gamma(U, \Phi(\mathcal{F}))=\prod_{x\in U} \mathcal{F}_x$$ with the obvious restriction maps. (This is the same as endowing the etale space of $\mathcal{F}$ with the trivial topology and considering the sheaf of sections.) There is a natural injection $\mathcal{F}\to \Phi(\mathcal{F})$ sending a section of $\mathcal{F}$ to its stalks. Note that $\Phi$ is functorial; indeed, it is the right adjoint to the natural inclusion of the full subcategory (Flasque sheaves on $X$) $\hookrightarrow$ (Sheaves on $X$).
See for example this Wikipedia article or Godement's book on sheaf theory.