Single mother not married with fathers name only on birth cert. want to add mothers surname

How can the mother's name not be on the birth certificate? Do you mean your child's legal name is the father's?

My son has is father's surname but I want him to have my surname too. Not necessarily to use a double-barrel name but just that he has the option to use either in later life as we have split up. My older children have my surname

Sorry, no way to 'have a option" he will have what ever name is on the birth certificate, you can with fathers permission add yours with a hyphen and then that will always be his last name legally