The novelist Mary Antin, a Russian-Jewish immigrant to Boston, recalls that soon after her arrival, her friends and family picked a new 'American' name for her.
When most immigrants changed their first name, it was usually to a name that shared only the same initial letter or sound, e.g. Someone named "Moshe" or "Mendel" or "Mordcha" in the old country might take the new American name "Max" or "Morris" or "Marvin".
There are no rules regarding these transformations; there is no one-to-one correspondence; the names don't "translate" or "mean" anything. People were free to choose whatever name seemed fashionable. These changes not officially recorded, for the most part -- the immigrant just started using their new 'American' name.