What is the ruling of Islamic law regarding a person telling his mother that the Messenger of Allah, may Allah's prayers and peace be upon him, and Abu Bakr took her deceased brother to alleviate her grief? And is expiation required for that?

The act of lying you committed is forbidden; indeed, it is one of the gravest slanders, as mentioned in the Hadith. There are other ways to console your mother besides lying, such as reminding her of the reward for the patient, and that grieving does not bring back what is lost, and the stories of the patient ones, and the calamity of the Muslims with the passing of the Messenger of Allah, peace and blessings be upon him. You must repent to Allah Almighty for this lie, and there is no expiation for you except repentance.