Don't add your mom's name to your gift. Your mom wants to hate on her FDIL, let her... and let her suffer the consequences of doing so. You on the other hand seem to want to welcome your SIL into the family, and you should receive the recognition for that.
As for who's right... well for sure I think you mom is wrong and rude (and maybe nasty) and even if she isn't family, just a girlfriend or even just a neighbor or coworker or roommate, I think its pretty awful to have someone over for Christmas and purposely not have a gift for them! Even just a cute pair of snowman socks or small umbrella or blank notebook, IMO you give everyone something.