I legally changed my daughter's surname - as far as the state of New York is concerned.
Is it necessary, by " law " to change the surname on her birth certificate as well? I don't recall being told this needed to be done also. For the record, I want it to stay the name I gave her at birth. Someone at Social Security is under the impression this is needed to show proof I changed the name in order to get her new SS card. Thank You.

You just need paperwork to show a name change. But if you want you could also hyphnate the BC so both names appear.