There are already some excellent answers here, but let me add something from a slightly different point-of-view.
First of all, homomorphism is not a word that can be used nakedly (so to speak) in mathematics. It is a function between two structures that preserves their structure, in some sense, and this sense has to be specified as part of the theory of those particular structures. (E.g. in the theory of groups, we define group homomorphisms; in the theory of rings, we define ring homomorphisms, etc.)
Also, in certain contexts, there can be competing notions of homomorphisms (for rings with identity, should the homomorphisms preserve the identity?; for Banach spaces, should morphisms be simply continuous, or should they preserve the norm?); in any particular situation, if it hasn't already been made clear in the context, you have to specify which definition you're using.
Now, in some situations, we don't use the word homomorphism; e.g. we don't speak of a homomorphism of topological spaces (instead, we have continuous maps between topological spaces). The choice of whether or not to use the word homomorphism is to some extent dictated by tradition, and also by general mathematical culture; the more algebraic the structures under consideration are, the more likely the word homomorphism is to be used in regard to them.
Now category theory abstracts the notion of mathematical structure and structure preserving maps between them. As already noted in other answers, not all categories are concrete, or, even if they are, are not naturally thought of as being concrete. And if we're talking about an arbitrary category, we don't specify what sort of objects the objects actually are; we just write $\mathcal C$, and $x \in\operatorname{Ob}\, \mathcal C$, and $f: x \to y$, without specifying what $\mathcal C$ is (just as in the theorems of a group theory text, we don't specify what the group $G$ is or what it's elements $x$ are or what its operation is; it is just some abstract, unspecified group).
Now the founders of category theory knew that they were abstracting a situation in which very often the category was going to consist of algebraic structures and homomorphisms between them. But they also knew that they were abstracting examples like topological spaces and continuous maps, where homomorphism is not the traditional term. Hence, they chose morphism as being close enough to homomorphism to be suggestive, but not identical, so that it didn't give the psychological impression of ruling out contexts like topological spaces and continuous maps. [This last paragraph is my invention, not actual history, but is probably fairly close to the actual history, and hopefully gets the point across.]
Finally, it seems worth noting that often now in mathematics people speak of a morphism of rings or groups or ..., i.e. they drop the prefix homo-, and I think this is the influence of the general category-theoretic terminology.