Who has the greater right to name the newborn: its mother or its paternal grandmother, especially when the father has equal options and the matter reaches the point of dispute?

Naming is the father's right, and if the parents disagree, the matter reverts to him. The most beloved names to Allah are "Abdullah" and "Abdul-Rahman." The father may relinquish his right to name the child to the mother. It is preferable for him to name the child with the name the mother desires if that pleases her, due to the obligation to obey her in what is good and involves no sin.