Among both Sephardic and Ashkenazic Jews, there is a custom to name a child after someone, usually a family member, who has died. The usual explanation for this practice is that the parents hope that in receiving the name of an admired family member, the child will emulate in life the virtues of the deceased.
Ashkenazic Jews (Jews of Central and Eastern European origin), Sephardic Jews (Jews of Iberian or Middle-Eastern origin)
To a certain extent, too, it is believed that the soul of the loved one lives on in the child who now bears his name. Indeed, learning about the people for whom they are named is an excellent way for children to identify with the history of their own Jewish families and, by extension, with the history of the whole Jewish people.
Some parents choose a name that has the same meaning — or a similar meaning — as the name of the relative they are naming after. For example, baby Zev might be named after Velvel (both mean wolf, but the first is Hebrew and the second is Yiddish). Or Leora — a modern Israeli name for girls meaning “my light”–might be named after Meir — a more traditional man’s name that means “one who illuminates.”
Some parents choose a name that sounds like the name of the relative they are naming after. Some people try to find a name with a whole consonant cluster that is phonically similar (like Amalia for Emily), and some just choose a name that starts with the same letter (like Matan for Menachem). This can of course also extend to names that are not strictly-speaking Jewish names. Many Jewish Samanthas are named after a Samuel or a Shmuel.
While you might be happy to give your daughter the Yiddish name Shaindel, which belonged to your beloved grandmother, some parents today choose to modernize, or at least tweak, the original name of the person they’re honoring.