She vowed to beat her son

You have to offer kafaarat yameen (expiation for breaking a vow), because this beating is not an act that can bring you closer to Allaah; it is the matter of a personal attempt to deal with something. So if you did not do it, you have to offer kafaarat yameen. And because beating him until his blood flows is not permitted.

The vow which you describe in this case is like a vow to commit sin; it is not permissible to fulfil a vow to commit sin, and the expiation for that is kafaarat al-yameen, which is:

Feeding ten poor people, or clothing them, or freeing a believing slave. Whoever is unable to do any of these three things can fast for three days.

Feeding the poor is done by giving them half a saa’ of the staple food of the country, whether it is dates, wheat, rice or whatever. The equivalent weight is approximately one and a half kilograms.

And Allaah is the source of strength.