How to find the smallest flabby sheaf containing a given sheaf ?

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 $$Gamma(U, Phi(mathcal{F}))=prod_{xin U} mathcal{F}_x$$
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.

Edit: One can also mimic your construction of $C^k_{nd}$ as follows. Let $mathcal{F}_{nd}$ have global sections given by $bigcup Gamma(X-partial U, mathcal{F})/sim$ where the union is taken over all open sets $U$, and we say $(f, X-partial U)sim (f', X-partial U')$ if $f=f'$ when restricted to $X-(partial Ucup partial U')$. Then local sections will be restrictions of these global sections. This will always be flabby, but will not necessarily have a morphism $mathcal{F}to mathcal{F}_{nd}$ unless $mathcal{F}$ has enough sections (for example, if $mathcal{F}$ is fine, as in your example).