What name should be given to the new-born after he was named Mu'adh, then 'Umar, then a vow was made to name him Muhammad?

If you vow to name your son "Muhammad" as an act of drawing closer to Allah through the name of His Prophet, then you must fulfill the vow and name him Muhammad. This is because fulfilling a vow of obedience to Allah is obligatory, and naming children after prophets is recommended, as the Prophet, peace and blessings be upon him, said: "Whoever vows to obey Allah, let him obey Him."

However, if you did not intend to name him after the Prophet but rather liked the name for its beauty or because it matched the name of a beloved person, then you are not obliged to fulfill the vow. This is because a vow is only binding for recommended acts (mandubat) and does not include permissible acts (mubahat), obligatory acts (wajibats), forbidden acts (muharramat), or disliked acts (makruhat).