Changing your name on your Social Security card is free. To start, fill out the Application for a Social Security Card. Submit additional documents to prove your age, identity, citizenship, or immigration status. In general, these are the following documents to submit with your application:
A birth certificate to prove your age
One of the following to prove your identity: U.S. Driver's License, U.S. state-issued non-driver identity card, or U.S. passport
Your U.S. birth certificate or passport to prove your citizenship
A current unexpired document issued to you by the Department of Homeland Security showing your immigration status, such as Form I-551, Form I-94, or Form I-766