Is the money that the mother forbade herself (from taking) considered forbidden for her after her son returned it to her as a gift?

Her mother's mere statement that the money is forbidden to her, without taking an oath, does not make it forbidden for her. Not every prohibition necessitates an expiatory offering for an oath. Allah condemned those who forbade what is lawful, yet He did not obligate an expiatory offering upon them. Therefore, it is permissible for your mother to take the money from your brother, whether as a return for what she gave or as a gift.