Board Chair Hilda Solis introduced the motion for the reward on Tuesday.
Galvan's brother Larry Galvan, 27, was fatally shot on Monday outside their mother's home in east Los Angeles.
A second brother, Dennis, was wounded in the foot and is expected to fully recover.
Sheriff's investigators are calling it a targeted gang shooting, but the councilman denies that.
He believes it was a case of mistaken identity. He is calling on the unidentified gunman to surrender.
The councilman released a statement on his Facebook page on Monday.
A makeshift memorial was set up at the scene of the crime. Family members said Larry Galvan left behind a loving wife and three young children.
The investigation was ongoing. Anyone with information on the incident was encouraged to contact the Los Angeles County Sheriff's Department's Homicide Bureau at (323) 890-5500.