The coroner's office says Hall, 32, died of a single gunshot wound and was "shot by a known assailant."
The Riverside Press-Enterprise reports police were called to Hall's home at 4:04 a.m. Sunday.
There's no information about the child suspected in the shooting. Hall's wife and five children were home at the time. Lt. Ed Blevins says it's unclear if the shooting was intentional or accidental.
The local leader of the National Socialist Movement lost his November bid to replace incumbent Tom Evans on the Western Riverside Municipal Water District board.