In New Zealand, you can choose to give your child your surname, the fatherâs surname, or any other surname you want-as long as it is an acceptable name (see the previous question). It doesnât have the same legal implications of paternity that having the fatherâs name on the childâs birth certificate would.n New Zealand, you can choose to give your child your surname, the fatherâs surname, or any other surname you want-as long as it is an acceptable name (see the previous question). It doesnât have the same legal implications of paternity that having the fatherâs name on the childâs birth certificate would.