I'm not sure what you mean by "the old way of naming children was to list their given name then the parent's given name". I've never heard of this as a custom; can you point to a source? In Ashkenazic practice, there is an extremely strong custom of NOT naming a child after a living parent.
It's more likely that Itte Beila was given two names, possibly in honor of two different deceased relatives. She may have preferred to use Beila, and later Americanized it to Bertha. (Itte is a Yiddish version of Yehudit/Judith.)