IF he had an original birth certificate (doubtful) then it would be amended to reflect the name change. But here's the thing...under Hawaii state law it would have to have "Amended" stamped in big letters for all to see.
But I don't think there was an original. Where ever he was born I think they took advantage of Hawaii's rather liberal laws at the time and wanted Obama to have US citizenship so they claimed he was born in Hawaii and provided apparently sufficient "evidence" of this and thus the state of Hawaii has deemed that he was born in Hawaii in spite of there being no birth certificate.
So I wonder if this also has alot to do with shuttling young Barry between mom and the grandparents.
Maybe when he was sent to live with them, grandma and grandpa messed with the records in Hawaii somehow?