"But they are Jewish and from what she tells me you are not allowed to name someone after someone still alive.In the story of John the Baptist's birth (see Luke) the relatives assumed that the boy would be named Zachariah, after his dad. After Z wrote "his name is John" on a tablet, he could speak again. Admittedly, this is the New Testament, but I assume Zachariah was a law-abiding Jew.
It is a custom of Ashkenaz Jews not to name a baby for someone still alive but Sefardic Jews do not have this custom and it is not uncommon to find, among Sefardim, father and son with the same name.