James Cho died alongside his parents Cho Kyu Song, 37, and Kang Shin Young, 35, according to reports. His six-year-old brother was injured but survived.
BBC reports the identifications come as officials probe whether the killer had links to any far-right organisations or beliefs.
Eight were killed in the shooting.
Several victims remain in hospital.
A verified GoFundMe page says that the Cho family were at the Allen Premium Outlets mall on Saturday to exchange clothing their six-year-old son had received as a birthday gift a few days before.