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.
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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.