Here's one example of the Americanization of a Yiddish given name. My cousin told me that her mother, my great-great-aunt "Becky", came to New York on a particular ship from Hamburg in April 1907. When I found that ship list, the given name in the entry read "Scheine Rywka". After speaking to my cousin again, she assured me that this entry was indeed for her mother, whose Yiddish name was "Sheina Rivka".

"Sheina Rivka" is an unrelated double name pair, and because this was a German ship, the German clerk wrote the name down using German orthography (spelling) as "Scheine Rywka". After immigration, she chose the new 'American' name "Becky", because "Rebecca" is the English language version of one of her Yiddish names, "Rivka", and "Becky" is the English diminutive of "Rebecca".

So while there is some logic to this Sheina Rivka's choice of a new American name, this is not always the case...