A 60 day extension based on visiting family could be gotten if you had your name on the birth certificate. And you could get a 90 day single entry Non-O visa from the thai consulates in Vientiane or Savannakhet with the same documentation.
However you could NOT get a yearly extension of stay to take care of a half-thai child just with your name on the birth certificate. That is when you'd need to do what
@Pa**
, suggested and go thru family court to get recognized as the legal father and granted parental rights. That's because ALL parental rights with children born out of wedlock are granted to the mother.