The camino goes straight through the heart of Oviedo, from the Cathedral almost to the train/bus station, so any place centrally located will be near the camino. In the old town, I like Hostal Arcos, though its prices seem to have risen. The Vetusta is another place I like a lot, I think it is a two-star hotel. I have been in Oviedo a number of times on different caminos, and what I usually do is just head to booking and am usually able to get a last minute great deal.
Here is a screen shot of the area between cathedral and train station, so you can compare with a map on booking if you just take pot luck with a last minute rate and want to be close to the camino. I just looked for tomorrow and the Reconquista has a room for about $100 — though that’s clearly not a budget rate, it is quite a good rate for that place, which is where some of the Penelope Cruz movie Vicky Cristina Barcelona was filmed.
Pensión Romero is another option — I’ve never stayed there but think other forum members have recommended it.
You will love Oviedo, and if you are a fan of ancient architecture, the Naranco sites are just spectacular.