Must my mother make up the fast day that she intended to fast but did not fast after the time of Fajr began?

If the fast your mother intended was a supererogatory (nafl) one, then it is not obligatory for her to observe it, and she is permitted to break it, based on the Prophet's (peace and blessings of Allah be upon him) saying: "A volunteer fasting person is the master of himself; if he wishes, he fasts, and if he wishes, he breaks his fast." However, if the fast was obligatory, such as making up for missed Ramadan days or a vowed (nadhr) fast, then it remains a liability upon her. There is no sin upon her for revoking the intention to fast, as long as she has not yet started the fast. But if she intended to observe a make-up fast and then broke it after dawn had broken, she has committed something unlawful, and it is recommended for her to make up for that day.