Is expiation incumbent upon the son or the mother if the son swore that the mother would not return the gift and the mother swore that she would not take it, and then the mother gave the gift in charity to a needy person?

If the mother swore not to take the money you gifted her, and then she took it: if her intention was not to take it for personal benefit only, and she then donated it as charity, there is no breaking of the oath upon her, because oaths are based on intention. However, if she did not have a specific intention, then she breaks her oath by taking the money and donating it as charity, and the expiation for breaking an oath becomes incumbent upon her. As for you, there is no breaking of an oath upon you as long as the money was not returned to you.