Yes, the father will have a right to object to the application for a name change. It will be up to a court to allow the change or not but given the children's objections to changing it to the mothers married name, there is a good chance it won't get changed to that. There is no real reason to change it to the mothers maiden name since no parent uses that name.
In some states they like to attempt to appease both parents and allow a hyphenated name but realistically I would not want to burden my children with that issue.

