There is no law but only custom regarding granting a Hebrew name. As is well known, Jews descended from Eastern European Jews tend to name children after deceased relatives; Sephardic Jews often name children after living elders. Both approaches are attempts to foster honor, memory and love.
Acquiring a Hebrew name has been a spiritual as well as practical tactic for Jewish survival through centuries of diaspora. Like all arrivals to a new country, Jews acculturated by adopting names from native languages, whether Polish, French, English, etc. But Jews also maintained the custom of calling each other by Hebrew or Yiddish names, especially in ritual matters, such as when called to read from the Torah, when married, when mentioned in healing prayers, and when their names were inscribed on grave stones. Sometimes Jews called each other by both their adopted names and their Hebrew names. A vestige of this practice is heard in names like Dov Be’ar or Ze’ev Wolf.
There are many critieria for picking a name for a youngster or an adult, other than naming after a relative. First, one might pick a Hebrew name merely because one finds it sounds pleasing.
One with a Hebrew-derived English name, such as David, Samuel, Evan, Ethan, Jake, Josh or Jonathan, has the option of taking the Hebrew equivalent: Daveed, Shmu’el, Evan, Eitan, Yaakov, Yehoshua or Yonatan. A Rebecca might like Rivka, or a Lisa or Liz might consider Elisheva, and Leah is Leah.
One can choose a name cued by similar consonants, such as a Susan taking the name Sara, or a Paul taking the name Paltiel, or Allen choosing the name Elchanan. A Brianna might like Bracha.
Someone who likes music might like the name Rina or Shira, both meaning “song.” Someone who likes the theme of light might take the name Ori, Nogah or Shachar (“dawn”).
Someone born on or near Passover might be named Pesach. Boys or girls born in the Hebrew months of Adar, Nissan or Sivan might acquire the name of the month in which they were born. Aviv or Aviva might be given to a child who arrives in the spring.
The name Oz (rhyming with “grows”) means strength. Emunah means faith or faithfulness. Emet is the Hebrew word for truth. Doron means a gift. Bracha is Hebrew for blessing. Noach or Menachem connote a person who gives comfort. Simchah translates to joy or rejoicing. Noam means pleasantness. Avi or Abba literally mean father in Hebrew and are very popular names in Israel. But in Jewish tradition, a teacher is also called father, and these names connote knowledge and wisdom as well as maturity.
Thus far we have discussed only first names. When used for ritual purposes, such as being called to the Torah, a Hebrew first name is followed by the phrase ben or bat (“son…” or “daughter of…”) and then the Hebrew first names of a father and/or mother. For example, David son of Jesse would be “Daveed ben Yishai.” In egalitarian Jewish settings the name of the mother is included: Adar Shoshan ben Elchanan v’Esther Leah. If the father is a descendant of the Levites or Kohanim, the father’s name is rendered: Moshe ben Amram ha-Levi v’Yocheved. Eliezer ben Aharon ha-Kohen.
Two final notes: Modern Israeli Hebrew renders the word “daughter” as bat though many Jews still use the European pronunciation “bas” as in “bas mitzvah” rather than “bat mitzvah.”
When one utters a Hebrew name for a healing prayer in the synagogue, it is customary to identify the ill person by citing the mother’s name without the father’s. If Harry is sick, a family member might mention his Hebrew name as “Hayyim ben Devorah.” Among the reasons for doing is this is the possibility of eliciting more compassion from heaven if his or her mother is mentioned alone. This practice might also arise from a superstitious attempt to throw the Angel of Death off track by referring to an ill person with a slightly different identity.