Do Jews name their kids JR's.
JR Smith is from Lakewood, but he's not Jewish.
In all seriousness, IINM, Ashkenazi custom is not to give the name of a living family (father, mother grandfather, grandmother, etc. Uncles, aunts and cousins are ok). Sefardic custom is to name after living grandparents. But even then, they would just give the name, no "junior".