What is the ruling on a non-Muslim mother holding a birthday party for her Muslim son? And how can the son refuse that without offending his mother?

Every recurring Eid, whether weekly or annually, that is not legislated is considered an innovation (bid'ah), such as Christmas. For in Islam, there are no Eids except Eid al-Fitr, Eid al-Adha, and Jumu'ah (Friday). This is not considered a custom (aadah) because it recurs.

As for how to deal with the mother, one must be frank with her that this act is not permissible شرعاً (religiously), and that participation in it is not possible. She should be informed of this in the best manner. If she is not convinced, then one must ensure to be out of the house during the time of the celebration.