If you have a child who, through a dna test, was found to have a different father than the one on the birth certificate, how can you change the name on the birth certificate?
See this link: http://www.cdph.ca.gov/certlic/birthdeathmar/Documents/Affidavit-BirthPAMPHLET-(11-10)-MERGED.pdf
It will require a court order. That can be done through a petition for parentage.
Your ability to change the name on the birth certificate or take your name is also affected by the age of the child and the mother's willingness to change the name.
A parentage action is required, which must be filed within 2 years of the birth of the child.