One often hears "The name was changed at Ellis Island".   No, it was not, despite the popular myth and a few good jokes.   Many, if not most, Jewish immigrants to the United States (and Israel, Latin America and elsewhere) did eventually change both their given names and surnames, but this came later, as part of the assimilation process. They adopted new names after they began working or started school. Often, it wasn't even the immigrant who invented their new name; it might have been a Jewish neighbor or an Italian co-worker or an Irish schoolteacher...

Moreover, it cannot be repeated too often that there are no fixed English equivalents for immigrant Hebrew or Yiddish names. Immigrants were free to choose any new name they wanted -- there were no rules or regulations.