There are no rules in the Jewish tradition for how to name a child. Jews from different parts of the world follow different, sometimes contrary practices.
Jews of Central or Eastern European descent (Ashkenazi Jews) compose the majority of the Jewish population of the U.S. Ashkenazi tradition strongly discourages naming children after close living relatives, especially the parents, but encourages the naming of children to honor dead relatives, often grandparents. Often the child will be given a name that starts with the same letter as an honored late relative.
Jews who trace their ancestry to Spain, Portugal and North Africa (Sephardic Jews), however, may name children for living relatives.
It is also customary–and an essential part of the brit milah ceremony–to give a Jewish child a Hebrew name in addition to an English name. Sometimes the Hebrew name is the same as a relative’s Hebrew name, sometimes it is simply related to a relative’s name, sometimes it is the Hebrew form of the English name (e.g., Yosef for Joseph, Dovid for David). Many parents choose their children’s name from the Bible, from rabbinic literature or from the pool of modern Israeli names.