See here for a few examples, such as recursion theory becoming computability theory, (linear) complex groups becoming symplectic groups (change due to Weyl), bicompact becoming compact, and pre-schemes becoming schemes.
Other examples: Gaussian integers used to be called complex integers, the term "group" initially referred only to what we'd call finite groups (existence of an inverse was not initially part of the definition, since it followed from the definition using finiteness) and Lie, Lie algebras were previously called infinitesimal groups (the change to Lie algebras is due to Weyl), cyclic algebras were initially called "algebras of type D" by Dickson, tensors were called "systems" by Ricci and Levi-Civita and "direct products" by Murray and von Neumann, topology was called analysis situs (change due to Lefschetz), and commutative algebra was initially called ideal theory.
The meaning of homomorphism and isomorphism did not settle down for a while: in Jordan's 1870 textbook on group theory (a subject that for a long time was called the "theory of substitutions" using the old-fashioned term substitution for what we'd call a permutation), he used isomorphism for what we'd call a group homomorphism and holoedric isomorphism for what we'd call a group isomorphism (the term simple isomorphism was also used for modern-day isomorphisms). A discussion of this is here. Normal subgroups used to be called invariant subgroups.