"North California" and "South California" sound odd. Nobody calls them that. It ought to be "Northern California" and "Southern California".
Central California is pretty much all agriculture, so you could pick just about any fruit or vegetable name and be safe. From what I've read, CA produces almost all of the nation's artichokes, walnuts, garlic and almonds, so those are big ones. Oh, and as far as the LA metro area needing a concise team name, as far as I can tell, they don't care. I'm partial to the "Los Angeles Southern Californians of North America", myself.
