Let $k$ be a field, $\ell$ a prime different from the characteristic.
If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local system on the henselization of $Y$ along $S$ comes from a $\mathbb{Z}_{\ell}$-local system on $S$?
I think you mean to ask if a $\mathbf{Z}_{\ell}$ local system on $S$ comes by reduction of a unique one on the henselization of $Y$ along $S$ (which isn't quite what is written, but is likely the intent in view of the special case when $S$ is a point).
This is affirmative for any affine scheme $Y$. The main task is really to prove for the constant group scheme $G = {\rm{GL}}_n(\mathbf{Z}/N\mathbf{Z})$ that every $G$-torsor over $S$ lifts uniquely up to unique isomorphism to one over the henselization of $Y$ along $S$. (It is not necessary to assume $N$ is a unit on $S$.) The strong uniqueness aspects would then allow one to vary $N$ through powers of a prime $\ell$ to uniquely lift a lisse $\mathbf{Z}_{\ell}$-sheaf in a similar manner.
The short paper "Principal Homogeneous Spaces over Hensel Rings" by R. Strano in Vol. 87 of Proceedings of the AMS, pp. 208--212 (1983) proves that if $Y$ is affine with henselization $Y'$ along $S$ then the map of sets of isomorphism classes of torsors for the etale topology ${\rm{H}}^1(Y', G) \rightarrow {\rm{H}}^1(S, G_S)$ is bijective for any smooth affine $Y'$-group scheme $G$. Thus, $G$-torsors on $S$ lift uniquely up to isomorphism to $G$-torsors on $Y'$.
It remains just to check the rigidity aspect when $G$ is finite etale : for such $G$, if $E$ is a right $G$-torsor on $Y'$ then is ${\rm{Aut}}_{Y',G}(E) \rightarrow {\rm{Aut}}_{S,G}(E_S)$ bijective? The functor of $G$-equivariant automorphisms of a right $G$-torsor is itself represented by an etale-twisted form of $G$, so by a finite etale (group) scheme $A$ over $Y'$. Thus, it reduces to the question of bijectivity of $A(Y') \rightarrow A(S)$ with $A$ finite etale over $Y'$. This in turn is part of the content of $(Y', S)$ being a henselian pair.
