Casa Florentina, in the very village of Arenas de Cabrales, is run by a very kindly young couple. The low season rates are very low: 25 € a double!
Unfortunately, as we'd booked late, we could only stay one night.
No problem, though; we walked around the corner and saw the Hotel Mestas. (Next picture)
It had a room available for 5 nights, and cost only about 10 € more. It was very well-kept and comfortable.