It does take a court order to change a child's name. You can file a Motion in the divorce case, requesting the name change. Once the order is granted, you then have to have the birth certificate amended.
As an adult, you can change your name without a court order, but my experience has been, that many institutions such as banks, credit cards, social security, do not recognize the name change, without a court order. If you as an adult want to change your name, you can go to your local courthouse and file a Petition for Name Change.
A weekly guide with tips and legal advice for each stage of the process.