The practice among Sephardim is to name children after their grandparents even if the grandparents are still alive. This is in contrast to the custom of the Ashkenazim, who make a point of naming children after grandparents (or other relatives) only after their passing.
The Sephardic custom has ample precedent in ancient tradition. One of the great Tanna'im (sages of the Mishna), Rabban Shimon Ben Gamliel, named his son Gamliel – his father's name – during his father's lifetime (see Seder Hadorot.) Even earlier in history, Kalev Ben Yefuneh, a personality mentioned in the Torah, begot a son named Chur, who named his son Kalev even though the grandfather was still living. (Rashi, Divre Hayamin I, 2:50.)
Another custom practiced by the Sephardim is to name the first child after the husband's parent, and the second child after the wife's parent. This, too, has precedent in the Torah. The Torah tells that when Yehuda's first son was born, "He named him Er" (Bereishit 38:3), but after the birth of his second son, the verse writes, "She named him Onan" (Bereishit 38:4). This suggests that Yehuda chose the first child's name, whereas his wife chose the second son's name. (Da’at Z’kenim, Parashat Vaysehev.) Furthermore, it has been suggested that Aharon's failure to observe this practice when naming his two oldest sons, Nadav and Avihu, may have contributed to their tragic death at an early age. Aharon named his first son Nadav, after his father-in-law, Aminadav, whereas his second son he named Avihu, which means "He is my father," in honor of his father. He thus deviated from the practice of naming the first child after the father's parent, and some Rabbis saw this as a contributing factor that led to Nadav and Avihu's tragic fate.
Therefore, parents should ensure to follow this practice, and name the first child after the father's parent, and the second child after the mother's parent. If, however, the father's parent foregoes on the honor and allows the first child to be named after the maternal grandparent, then the parents may name the child after the maternal grandparent.
The verse in Mishlei (17:6) states, "Ateret Zekeinim Benei Banim" – "Grandchildren are the crown of grandparents." Grandparents receive much pride and joy from seeing their grandchildren, and when they see the grandchildren carrying and perpetuating their name, this is particularly gratifying. Parents should therefore follow this custom to name their children after their grandparents even during their lifetime.
Summary: The practice among Sephardim is to name children after grandparents even if the grandparents are still living; the first child is named after the father's parent, and the second child is named after the wife's parent, unless the father's parent forgoes on this honor. Parents should ensure to follow this time-honored tradition, which has its roots already in the Torah.
See Soba Semahot, Helek 2, page 82.
