The fossil beds spread out over a large area, so I've stayed in a couple of places in the area. Clyde Holiday is one, and another is in the town of Fossil at the Wheeler County Park near the rodeo grounds. Inexpensive, with full hookups for $18.75 per night (weekly rate). Fossil has an interesting town museum with the only gasoline power iron I've seen.
When in the area, be sure to see the Kam Wah Chung & Co building in John Day. Another interesting museum or sorts.
Some photos.