Knowing how procedural Social Security is and requires their people to be, changing your name on your birth certificate to match your SS card, as Janke suggested, may be the easiest way to go without incurring significant legal costs. You will need to check on the requirements to have an amended birth certificate issued in the state you were born to see if this is a viable option for you. It will undoubably require a court order but should not be a costly process. If you post the name of the state, someone here may know what it will take.

