My son was born 12 years ago. His father was not put on the certificate as he wanted nothing to do with us. Now he wants his name on the certificate. Can he get his name put on? Can he do it without my say so?

He can petition the court and ask to have it added. I don't know why. Has he established paternity though the courts using DNA yet ? Has he filed in court for visitation or even joint custody yet.

Unless he has established a relationship with the kids yet, and proven paternity I doubt he will get that at this point, He can file for visits and get that.

Has he been paying his court ordered support yet ( I assume you at least got a court order for child support?)

Can he? Yes. If he files a petition with the appropriate court and they decide to grant it. Will they? Its unlikely if he has just come out of the woodwork to claim his son. But if he decides he wants to be a father now and proves to the court he will be, then yes they may very well grant it.

ANY question on law needs to include your general locale as laws vary by area. And it is a pertinent question to what you have done over the last 12 years. It was NOT his decision to be left off the birth certificate. You could have listed him while still giving your child whatever last name you wanted. Have you done anything to promote or inhibit his being a part of his son's life? Have you filed for custody or child support? Did you offer visitation?