I am not completely clear, is the current GC in the maiden name or married name? It sounds like the 2 yr GC is the maiden name, and you are trying to change that via the ROC process. At least to me, this is a new one if my assumption is correct. Others may know better, but can one request a name change at the ROC stage? Regardless, what @OldUser said is accurate, you need a form of ID (GC, US passport, etc.) to take to the SSA to be able to get the name changed on the SSN.
If the ROC process will not allow for the name change on the GC, you can do that during the N400 process.
Good Luck!