You were told wrong. The SSA doesn't legally change names PERIOD. The SSA has a list of things that they will accept to change the name in their records. A certificate of Naturalization can't even be used to change your OWN name, let alone someone else's. To change her name (or yours), you'll have to go to court. Information on minor name change here: http://www.courts.ca.gov/1052.htm

