You can petition the court to ADD your name to mom's (hyphenate), it is unlikely they will approve CHANGING the name fully. Mom is still mom and you can't erase her from existence.
i must politely disagree only b/c in NYS, with a court order the name can be changed to whatever the petitioner is asking {assuming the order is granted} my husband is an example.
His ex put ANOTHER man's name as the father on the B/C. my husband had to get a DNA test, order of filiation etc, he had to take it to the health dept first to show proof he was the father and they changed his sons last name completely to just his last name only. there is no law in NY that i am aware of that states that both names have to be used{could be wrong, dont quote me on this}
BUT....how old is the child? i wouldnt change the name only for the childs sake, the child is used to a certain name and is already going thru a lot it seems with the custody change.
love your child more than you hate the ex, wise words from another member on here....