Here is one way of addressing your question. Let $X$ be a scheme, $U$ an open subscheme of
$X$, and $Y$ its closed complement (endowed with the reduced structure, say). Let $i: Y \rightarrow X$ and $j: U \rightarrow X$ denote the inclusion maps. Then the pushforward functor $i_{\ast}$ determines a fully faithful embedding from the category of etale sheaves on $Y$ to the category of etale sheaves on X. The essential image of this embedding is the full subcategory spanned by those etale sheaves $\mathcal{F}$ on $X$ such that
$j^{\ast} \mathcal{F}$ is final (in other words, such that $\mathcal{F}(V)$ has a single object for every etale map $V \rightarrow X$ which factors through $U$). 

In topos-theoretic language, this says that $i$ induces a closed immersion of etale topoi, which is complementary to the open immersion of etale topoi determined by $j$. (The above discussion makes sense in an arbitrary topos, and for a topos of sheaves on a topological space it recovers the usual notion of closed embedding).

I don't know a reference for the above statement offhand, but my guess would be that you can find a discussion in SGA somewhere.