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HelloThis is Samuel. I suggest you can do the name change on your own without local legal representation. A name change request can be made in the County Court or Supreme Court of the county where you live. If you live in New York City, you can go to the New York City Civil Court or the Supreme Court in the county where you live.
The clerk of that court can provide you a packet of forms to file for the Name Change. You will need a certified Copy of the child's birth certificate. And since there is no father listed, then you do not need to serve the Father. You should be able to file and have the name changed pretty easily.
Here is a LINK you can review and they have forms for Child Name Change for NY State and also if you are living in the City.