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In characteristic zero, I believe the answer is "yes". More generally, if X $X$ and Y $Y$ are two smooth proper curves over algebraically closed fields K $K$ and L $L$ of characteristic zero, then X $X$ and Y $Y$ have equivalent etale topoi if and only if they have the same genus and card(Kcard($K$) = card(L)card($L$). The "only if" claim follows from the fact that the etale topos has both the etale fundamental group and the Zariski site as invariants (the first being determined by the locally finite constant objects and the second being determined by the lattice of subobjects of the final object). This is more of a remark; the question is about the "if" claim.

For that, choose a set S and bijections f: $f: S --> X(K) \to X(K)$ and g: $g: S --> Y(L)\to Y(L)$. Now, for every finite subset T $T$ of S, $S$, let a_T: $$a_T: Spec(K) --> M_{g,T} \to M_{g,T}$$ be the map classifying f(T) $f(T)$ inside X, $X$, and ditto b_T $b_T$ for g(T) $g(T)$ inside Y. $Y$. Furthermore let A_T $A_T$ be the Henselization of a_T, $a_T$, and ditto B_T $B_T$ of b_T. $b_T$. For T $T$ inside T' $T'$ we have maps A_{T'} --> A_T $A_{T'} \to A_T$ and B_{T'} --> B_T$B_{T'} \to B_T$, mapping compatibly to M_{g,T'} --> M_{g,T}$M_{g,T'} \to M_{g,T}$; we consider the inverse limit of the fiber products A_T $$A_T \times_{M_{g,T}} B_T B_T$$ along these maps. Since it is (eventually) a filtered limit of nonempty schemes along affine dominant maps, it has a point over some algebraically closed field F. $F$. Let Z $Z$ be the induced curve over F $F$ and $h:S --> Z(F) \to Z(F)$ be the induced injection.

Then I claim that the etale étale site of X $X$ is equivalent to the subcategory of the etale étale site of Z $Z$ consisting of those $U --> Z \to Z$ with either U $U$ empty or Z-im(U) $Z-im(U)$ inside h(S). $h(S)$. By symmetry this will prove our desired result.

Here is how that equivalence goes. For $U --> X \to X$ be etale (suppose nonempty), let X' $X'$ in X $X$ be an open subset of X $X$ for which the pullback U' --> X' $U' \to X'$ is finite (i.e. proper). Take $T = f^{-1}(X-X')f^{-1}(X-X')$, and recall we have maps Spec(K) --> $Spec(K) \to A_T --> M_{g,T} \to M_{g,T}$ where the composite classifies f(T) $f(T)$ inside X, $X$, and similarly Spec(F) --> $Spec(F) \to A_T --> M_{g,T} \to M_{g,T}$ classifying h(T) $h(T)$ inside Z. $Z$. Let C' $C'$ denote the pullback to A_T $A_T$ of the universal T-punctured $T$-punctured curve over M_{g,T}; $M_{g,T}$; it gives X' $X'$ over K $K$ (special fiber) and Z':=Z-h(T) $Z':=Z-h(T)$ over F $F$ (generic fiber). Since C' --> A_T $C' \to A_T$ is a smooth map with a smooth proper relative normal crossings compactification, the maps X' --> $X' \to C' <-- Z' \leftarrow Z'$ induce equivalences on categories of finite etale étale covers; thus we can transport U' --> X' $U' \to X'$ uniquely over to a V' --> Z'$V' \to Z'$, via some W' --> C'$W' \to C'$.

But in fact I claim that the same is true of $U --> X\to X$. Indeed, let C $C$ be the completed curve of C' $C'$ (over A_T), $A_T$), and for t $t$ in T $T$ with corresponding section c $c$ of C-C' $C-C'$ consider the completion of C $C$ along c, $c$, then subtract the section c $c$ to get $L --> A_T\to A_T$. On pullback to K $K$ this becomes L_x, $L_x$, Spec of a Laurent field at x=f(t) $x=f(t)$ in X, $X$, and ditto for F $F$ and L_z; $L_z$; furthermore once again the maps $L_x --> \to L <-- L_z \leftarrow L_z$ induce isomorphisms on categories of etale covers of L. $L$. On the other hand, the Galois type of the pullback of U' --> X' $U' \to L_x X'$ to $L_x$ completely determines the points at infinity of U' $U'$ lying above x $x$ (and ditto on the F $F$ side): they are the orbits of Galois (which is the profinite completion of Z$\mathbb Z$) over any geometric fiber. Thus we can add in points at infinity on both sides in a consistent and canonical manner.

This finishes the proof, I suppose: functoriality of this one-to-one correspondence (and its independence of X') $X'$) comes from us having made our constructions compatible with changing T $T$ and the fact that smooth proper compactifications of curves over fields are unique and functorial.

1) This proof is close to the proof I gave in the comments for K $K$ the complex numbers: the fact that all elliptic curves are homeomorphic is symptomatic of connected moduli, which was again a key fact in this proof (though in a hidden manner?)

2) However, the real surprise lies in the differences: to get an equivalence of etale &eactue;tale topoi one needn't choose a homeomorphism -- in fact any bijection will do! The field F $F$ whose existence was ensured above guarantees some measure of continuity no matter what the bijection. In fact I imagine that in higher dimensions the notions of being homeomorphic and having equivalent etale étale topoi diverges. For instance, I'll bet that $E x A^1 \times \mathbb A^1$ doesn't have the same etale étale topos as G_m x $\mathbb G_m \times \mathbb G_m$. Similarly, I wonder about two varieties occurring in the same proper smooth family having inequivalent etale étale topoi.
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In characteristic zero, I believe the answer is "yes". More generally, if X and Y are two smooth proper curves over algebraically closed fields K and L of characteristic zero, then X and Y have equivalent etale topoi if and only if they have the same genus and card(K) = card(L). The "only if" claim follows from the fact that the etale topos has both the etale fundamental group and the Zariski site as invariants (the first being determined by the locally finite constant objects and the second being determined by the lattice of subobjects of the final object). This is more of a remark; the question is about the "if" claim.

For that, choose a set S and bijections f: S --> X(K) and g: S --> Y(L). Now, for every finite subset T of S, let a_T: Spec(K) --> M_{g,T} be the map classifying f(T) inside X, and ditto b_T for g(T) inside Y. Furthermore let A_T be the Henselization of a_T, and ditto B_T of b_T. For T inside T' we have maps A_{T'} --> A_T and B_{T'} --> B_T, mapping compatibly to M_{g,T'} --> M_{g,T}; we consider the inverse limit of the fiber products A_T \times_{M_{g,T}} B_T along these maps. Since it is (eventually) a filtered limit of nonempty schemes along affine dominant maps, it has a point over some algebraically closed field F. Let Z be the induced curve over F and h:S --> Z(F) be the induced injection.

Then I claim that the etale site of X is equivalent to the subcategory of the etale site of Z consisting of those U --> Z with either U empty or Z-im(U) inside h(S). By symmetry this will prove our desired result.

Here is how that equivalence goes. For U --> X be etale (suppose nonempty), let X' in X be an open subset of X for which the pullback U' --> X' is finite (i.e. proper). Take T = f^{-1}(X-X'), and recall we have maps Spec(K) --> A_T --> M_{g,T} where the composite classifies f(T) inside X, and similarly Spec(F) --> A_T --> M_{g,T} classifying h(T) inside Z. Let C' denote the pullback to A_T of the universal T-punctured curve over M_{g,T}; it gives X' over K (special fiber) and Z':=Z-h(T) over F (generic fiber). Since C' --> A_T is a smooth map with a smooth proper relative normal crossings compactification, the maps X' --> C' <-- Z' induce equivalences on categories of finite etale covers; thus we can transport U' --> X' uniquely over to a V' --> Z', via some W' --> C'.

But in fact I claim that the same is true of U --> X. Indeed, let C be the completed curve of C' (over A_T), and for t in T with corresponding section c of C-C' consider the completion of C along c, then subtract the section c to get L --> A_T. On pullback to K this becomes L_x, Spec of a Laurent field at x=f(t) in X, and ditto for F and L_z; furthermore once again the maps L_x --> L <-- L_z induce isomorphisms on categories of etale covers of L. On the other hand, the Galois type of the pullback of U' --> X' to L_x completely determines the points at infinity of U' lying above x (and ditto on the F side): they are the orbits of Galois (which is the profinite completion of Z) over any geometric fiber. Thus we can add in points at infinity on both sides in a consistent and canonical manner.

This finishes the proof, I suppose: functoriality of this one-to-one correspondence (and its independence of X') comes from us having made our constructions compatible with changing T and the fact that smooth proper compactifications of curves over fields are unique and functorial.

Some remarks:

1) This proof is close to the proof I gave in the comments for K the complex numbers: the fact that all elliptic curves are homeomorphic is symptomatic of connected moduli, which was again a key fact in this proof (though in a hidden manner?)

2) However, the real surprise lies in the differences: to get an equivalence of etale topoi one needn't choose a homeomorphism -- in fact any bijection will do! The field F whose existence was ensured above guarantees some measure of continuity no matter what the bijection. In fact I imagine that in higher dimensions the notions of being homeomorphic and having equivalent etale topoi diverges. For instance, I'll bet that E x A^1 doesn't have the same etale topos as G_m x G_m. Similarly, I wonder about two varieties occurring in the same proper smooth family having inequivalent etale topoi.

3) The positive characteristic case is a bummer for me because of non-topological (i.e. "wild") phenomena and I don't have much to say.