I am a mother in California, where and how do I go about legally adding my last name to my 5 year old son's birth certificate? He only has his fathers last name at the moment and I will like for him to also have mine.

You have to go to court and get a name change. You will need the fathers permission. If he doesn't like that idea then you can go before a judge and request the name change and a judge will pass down a ruling if it can be changed or not. Usually its going to be with a hyphen in cases like this. With older children it can be denied completely.

I guess that raises the question, does the father give permission,