Q. Is it allowed to name a child after a living relative?
A. The Sephardim have no problem with this, but the Ashkenazim do. Historically the Sephardi usage was probably the more authentic, since the Talmud reports several cases of a son bearing his father’s name, e.g. Abba the son of Abba (B’rachot 18b).
Avoiding the name of a living relative may have some folkloristic implications such as not confusing the Angel of Death who may come after one Abba and instead catch another.
Some German Jewish communities got over the problem by giving a son a second name which was the same as his father’s first name. Thus Samson Raphael Hirsch was Samson the son of Raphael, and Nathan Marcus Adler was Nathan the son of Marcus.