Legal Question in Civil Litigation in California

She can but unless there is some specific througough defense inability to serve will provide, there is probably no reason to do so.

If your mother is the defendant but you were the one who was served, then the service may be invalid -- not because the form was filled out wrong but because it was handed to the wrong person. But your mother may have been served properly, depending on the circumstances. I would need more information before I could say.

As Mr. Selik notes, there may be little practical reason for your mother to challenge the service. If it was not valid, the plaintiff will probably be given another chance to do it correctly. Such challenges do make sense under some circumstances. I don't know if any of those circumstances are present here.

Your mother was probably subserved, which would make the summons correct. That means she is the defendant, and was served by them handing the summons to you, assuming you were at an address she was associated with, coupled with a copy of the summons and complaint sent by mail.

Mr. Roach is right. There is a provision in the service of summons statutes that allows a summons for a defendant to be served on someone else of at least 18 years of age at any known address for the defendant, coupled with mailing to that address. You still mark the notice to person served the same as if it was handed directly to them.