What is the ruling on naming one's son "Mu'awiyah" in support of him, and so that his family may love him and cease slandering him?

There is no objection to naming someone "Mu'awiyah" because the Prophet, peace and blessings be upon him, did not change this name, despite Mu'awiyah being a scribe of revelation. The claim that the name "Mu'awiyah" means "female dog" is a lie and a distortion of its linguistic meaning. "Al-Mu'awiyah" (with the definite article "al-") means "female dog," but without it, it means "strength of hand."

Mu'awiyah ibn Abi Sufyan was given the title "Khaal al-Mu'mineen" (Uncle of the Believers) because he was the brother of Umm Habibah, the Mother of the Believers. This title was affirmed by Imam Ahmad ibn Hanbal, who considered anyone who did not acknowledge it to be mistaken and deserving of ostracism until they repented.

Many Companions, Successors, and scholars were named with this name, which indicates its popularity and goodness.