@ Author
Your briefs are half!
However, your incomplete briefs beside the point follow below steps (applicable for American born children cases only);
Step 1
Since you are aware of city of Birth of your child. You will have to contact the U.S. Department of State and proceed according to Federal Laws. Otherwise, your application will be processed under the Law of the State that issued the birth certificate. [If you do not know the city, State officials may be able to find out for you].
Step 2
Research the Laws of the State in which the child was born. In many States, only the child or a parent named on the birth certificate may apply for a name change without a court order.
Step 3
Register with the State's putative father registry if one is maintained by the State of the child's birth and you are a biological father seeking to be named on the birth certificate. Be aware, however, that many States that maintain putative father registries have strict deadlines for Registration. Registration is considered evidence of paternity and not absolute proof.
Step 4
Assemble documentary evidence justifying the proposed name change. If you are the biological father, you may have to submit to DNA testing and you might need a Court order authorizing the name change. If you are a biological mother then some States ask for Court orders. If you are an adoptive parent, the adoption certificate will be necessary.
Step 5
Complete an application for amendment to the child's birth certificate, pay the filing fee and submit the application together with the filing fee and supporting documentation to the State's Department of Vital Records or the equivalent therein. The State will issue an amended birth certificate but will retain the original in its own records.
However, a false statement on a birth certificate amendment application can result in criminal penalties and USA is very strict in 'perjury' Laws unlike India where a client dies and his children fights for perjury cases left behind by dead parent(s) :-)