An argument that someone might make against the original claim to the argument is a counterargument. It means the person is objecting to something that was previously stated.
Counterclaim is a more specific term used in a court of law. It means the defendant files a lawsuit against the plaintiff similar to the one the plaintiff filed against the defendant. It could be compared to a counterattack in the sense that the response is similar to what caused it.
However, both terms are sometimes used as synonyms.