Looking for some place quiet that includes the golf cart for transportation around the island.
By far, the best place to stay in Avalon is Inn On Mt. Ada. This B&B was once the home of William Wrigley (who owned all of Catalina), sits on a hill overlooking the picturesque harbor. A fantastic place. They have golf cars on premises that will take you into town and you can use.
I think the best option in town is the Hotel Metropole. Right on the water on the main boardwalk, recently remodeled. Probably not your quietest option, but a fun place in a great location.