Is what the pregnant woman said – that she would name her male child after the Prophet Muhammad (peace be upon him) – considered a vow, knowing that she said this before the death of her brother, whose name her family wishes to give to the newborn?

If a woman utters a phrase indicating a vow, such as "Lillahi alayya" (Upon me for Allah) or "Alayya nadhr (definition)" (I have made a vow), naming the newborn Muhammad, then it is a contingent vow that must be fulfilled if a male is born. This is based on the Prophet Muhammad's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." However, if she does not utter a vow formula, she is not obliged to fulfill it, and it is permissible to name the newborn after her deceased brother, though it is preferable to name him Abdullah or Abdur-Rahman.