More than likely, immigrants’ names changed as part of the assimilation process. Perhaps your great-grandmother’s American teacher couldn’t pronounce her foreign name, so she gave the child an American equivalent. Or maybe your ancestors wanted to become more American or avoid prejudices, so they changed their names themselves.
My father shortened our Italian surname, DeBartolo, to Bart when I was born, so the family name would sound more “American.” Rarely will you find a legal name change. More often, the ancestor just assumed a new name. As long as he wasn’t doing it to be fraudulent, there was nothing wrong with adopting a new name. If you do suspect a legal name change, however, check court records in the county where your ancestor lived.