What should I do about my mother who refuses to change my son's name from "Musa" to "Layth" and threatens to disown me, and what is the ruling of Islam on changing a son's name from that of a prophet to "Layth"?

The son's name "Musa" should not be changed for two reasons: First, Musa is the name of a prophet, and naming children after prophets is desirable to keep their memory alive and to honor them. The Prophet, peace and blessings be upon him, named his son Ibrahim. Deviating from the name of a prophet contradicts showing respect to him.

Second is the mother's objection and her desire not to change the name. Obeying and honoring one's mother is obligatory, based on Allah's saying: "And your Lord has decreed that you worship none but Him. And to parents, do good." [Quran 17:23], and also based on the hadiths that prohibit disobedience to parents. It is obligatory to obey her in matters that do not involve disobedience [to Allah].