Article 16 of China's Marriage Law stipulates that children can take their father's surname or their mother's surname. Generally speaking, after the birth of a child, the name of the child is determined by consensus of both parents, so the change of the child's name should also be agreed by both parents. If parents divorce, either party has no right to change the child's name without authorization.
In accordance with the relevant spirit of the Reply of the Supreme People's Court on the Change of Children's Surnames (81 Famin Zi No. 11), the public security organ may refuse to accept the request of one of the two parties to divorce for changing the children's names without consultation or without reaching an agreement through consultation; If one party obtains the change of the child's name due to concealing the divorce fact from the public security organ, if the other party requests to restore the child's original name and the divorce parties fail to negotiate, the public security organ shall restore it. Therefore, if a child changes his/her family name after divorce, he/she has no right to change it without the consent of the other party. Even if he/she changes his/her family name, the other party also has the right to change it, and he/she will have to change it at that time.
To sum up, in this case, you'd better discuss with the child's father first. If he agrees, he can take the household register, ID card, the child's household register, divorce certificate and other materials to the local police station of the local public security bureau where the household register is located.