What was Kirito's actual family name?
Per the fan wiki, his surname was Narusaka (鳴坂) prior to his adoption.


  Born on October 7, 2008, Narusaka Kazuto was adopted by his aunt and uncle a year after his birth, when his biological parents died in an accident.


According to this news article and some other sources I found, this was revealed at the beginning of Unital Ring, originally in the January 2018 issue of Dengeki Bunko, which publishes SAO as a serialized story.