To change your name on your Social Security card:
You must show a recently issued document as proof of your legal name change. Documents the SSA may accept to prove a legal name change include:
If the document you provide for a legal name change does not give enough information to identify you or if you legally changed your name more than two years ago, then you also must show two identity documents including:
In addition to your name, these documents also must contain identifying information or a recent photograph.
If you are a U.S. citizen born outside the United States and SSA records do not show you are a citizen, you will need to provide proof of your U.S. citizenship. If you are not a U.S. citizen, SSA will ask to see your current immigration documents.
The new card will have the same number as your previous card, but will show your new name.
Social Security Handbook