Most people just started using a different name, whether it be first name and/or surname, and there is no legal paperwork.
Some people did use the naturalization process to change their name, and that would be noted on their nat papers, if they did so.
Some people did go to court and get their name legally changed.
It seems you are saying that you believe the surname the family used in The U.S. is not the surname they used before coming here?
Are you trying to figure out the original surname? If so, there are methods to attempt to do this, even if they didn't legally change the name.
Do you know what town they came from originally?
Good luck in your search,