The "obvious" notion of morphism between models of set theory is, of course, a function $\phi$ such that if $x \in y$, then $x\phi \in y\phi$. Is this the "appropriate" (or at least, a useful) notion of homomorphism?
This is indeed the right notion of homomorphism, models of set theory being after all a set with a binary relation.
That said, this notion itself does not seem to have been studied, and does not appear to be particularly useful. The variant that turns out to be very useful, in all kinds of set theoretic contexts, is the notion of elementary embedding. First of all, this is an embedding, so it is not just a homomorphism, but it is injective. Moreover, we require elementarity. This move separates the study of these maps from purely algebraic considerations.
The simplest elementarity we can require is for quantifier free formulas. The relevant maps $j:(M,\in^M)\to(N,\in^N)$ satisfy that they are injective, and for any $a,b\in M$, we have $a\in b$ iff $aj\in bj$. Actually, these maps have only been considered seriously very recently:
Joel Hamkins has proved, see here, that if $(M,\in^M)$ is a countable model of set theory, then there is such a $j:(M,\in^M)\to(L^M,\in^M\upharpoonright L^M)$. Here, $L^M$ is Gödel's constructible universe, from the point of view of $M$.
Most commonly, we require the maps $j:(M,\in^M)\to(N,\in^N)$ to be fully elementary, meaning that for any first-order statement $\phi(x_1,\dots,x_k)$ in the language of set theory, and any $a_1,\dots,a_k\in M$, we have $$ M\models\phi[a_1,\dots,a_k]\Longrightarrow N\models\phi[a_1j,\dots,a_kj]. $$ Note that in particular, $j$ is injective.
These maps appear naturally in the study of large cardinal notions, thanks to the ultrapower construction, the relation being that an ultrapower is well-founded iff the corresponding ultrafilter is $\sigma$-complete. That there are such non-principal ultrafilters is a large cardinal notion (measurability). That the ultrapower is well-founded provides us with strong reflection principles, which explains the usefulness of these embeddings.
The notion is weaker than being an isomorphism (in particular, we do not ask that $j$ is onto). This is an advantage, as there are no nontrivial isomorphisms between transitive structures.
In the context of fine-structure, it is useful to consider variants that are only partially elementary (we consider a whole hierarchy, depending on the complexity of the formulas involved; we also have different possible ways to measure complexity), and where the models $(M,\in^M)$ and $(N,\in^N)$ are not required to satisfy all of set theory, but only of a weak fragment, or they can even just be an initial segment of such models. These maps prove useful to carry out delicate recursive constructions, and are the reason behind the rich combinatorial structure (diamonds, square, etc) that $L$ and other fine-structural models carry.
If one is interested in these maps from a purely algebraic point of view, elementarity seems an unnecessarily strong restriction. Actually, it turns out that a significantly strong version of elementarity gives us a structure that is algebraically rich: Suppose that there is an ordinal $\lambda$ such that there is an elementary embedding (different from the identity) from the rank initial segment $V_\lambda$ to itself. (That there is such a $\lambda$ is a very strong large cardinal assumption.) The collection $\mathcal E_\lambda$ of all such embeddings admits two natural operations: Composition $\circ$ and application $\cdot$, and the structure $(\mathcal E_\lambda,\cdot)$ satisfies a left self-distributive law. Its study led to developments on braid theory, and much ongoing research. See for example the work of P. Dehornoy.