Why does the name "epimorphism" refer to a surjective homorphism? The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are very much welcome though).

In group theory, an epimorphism is a surjective homomorphism. Where does this name come from? Why have we chosen it?

As a comparison, a monomorphism is an injective homomorphism. Some authors like to call injective functions "one-to-one", and so one can see the rationale behind the "mono" part of the name. But for an epimorphism, I'm not really sure where the terminology has come from (despite reading the wikipedia section on it).
The prefix "epi-" in Greek has several meanings, but a common one is "upon, over". This is similar to the meaning of the prefix "sur-" in French, which was the origin of the term "surjective", introduced by Bourbaki. As such, both give the meaning that the function/morphism "covers" all of its range.
"Epic" has connotations of "long" or "all the way", as well as plenty of others consistent with being onto.

Think of an epic morphism.