In Ashkenazic practice (Jews of Eastern European origin), the custom is to name a child after a relative who has died, while Sephardim (Jews of Spanish and Middle Eastern origin) tend to name their children after living relatives. Reform practice allows either option. Jews-by-choice certainly may name their children after their parents, since Jewish tradition recognizes that all people, including people who have gone through conversion owe honor and respect to their parents.
The Eternal One spoke to Moses, saying, "Phinehas, son of Eleazar son of Aaron the priest, has turned back My wrath from the Israelites by displaying among them his passion for Me, so that I did not wipe out the Israelite people in My passion." - Numbers 25:10-11