I visited northern Spain this summer and can tell you about some places I especially enjoyed. I can only speculate on what the conditions will be like in January.
Barcelona is fabulous. Check historical weather to be sure, but I suspect it will be a bit more moderate than the Basque Country. For indoor sights you have a lot of museums and well as a significant number of modernista sights. I was there for 10 days and didn't have time for everything I wanted to see.
Girona has a lovely, large historic district with a number of indoor sights as well as the obvious strolling possibilities. It will not close for the winter. Definitely worth 2+ nights. Don't miss the art museum. There are side-trips possible to Figures and Cadaques, probably mainly for the Dali sites; Cadaques is a fishing-town-turned-beach resort, so I'm guessing it would be dead in the winter. I only changed buses in Figueres so can't comment on it.
There's lots of pretty, dramatic countryside in the Basque Country. It's much more interesting to travel through than the area around Madrid, for example.
My favorite side-trip from San Sebastian was to Hondarribia, a coastal town virtually on the French border. It's a place of some size, so buses should still be running pretty frequently in January (be careful about schedules on weekends, especially Sundays, though). Hondarribia has both a cute fishermen's quarter full of restaurants (don't know how many will be open in January) and an historic district on a hill. There's a good bit to see because you have the two areas of interest. This would be one of my top recommendations for a Basque Country side-trip.
On another day I used a combination of the slow, local FEVE train and buses to go to Zarautz, Getaria, and Zumaia. They are all attractive, historic coastal towns, fairly small. I don't know how much will be going on there in January. Zarautz, especially, is known for its wide, sandy beach, which would surely require a wetsuit in the winter.
Bilbao is worth decamping from San Sebastian for 2 or 3 nights, I think. You have the Guggenheim and the Bellas Artes Museum; I missed the latter because of a strike this summer, but that appears to be resolved. The good-sized historic district is attractive, and what I call the "downtown area" has lots of interesting, attractive early 20th century buildings. The city seems quite affluent, and I'm sure some advance planning will turn up interesting restaurant possibilities. You can definitely do a tapas crawl here. A sidetrip to the coastal town of Bermeo is worthwhile for a walk-around. I've mentioned 4 small coastal towns (Hondarribia's in a different category); unless you're running out of things to do, you don't need to try to see all of them, but I did enjoy them all.
I visited Viktoria-Gasteiz from Bilbao as a day-trip, but it's worth overnighting in. There's a large historic district (gotta love a city that calls its historic center the "medieval almond" because of its shape on a map) and two or three worthwhile museums. A feature of the historic area that I really appreciated was the outdoor escalators; they make it comparatively easy to get around that very hilly district.
A big "Yes!" to LaGuardia. Beautiful old hill town located near the La Rioja vineyards. I don't drink wine, but I noticed a lot of winebars in the town. Very atmospheric. Should be a nice place to spend the night. It's on the bus line from Viktoria-Gasteiz to Logrono, so it can be reached by public transportation. For some reason Rome2Rio.com doesn't know about that bus line. I probably learned about it by Googling.
I only changed buses in Logrono; I didn't have time to walk around. I gather from my guidebooks that there's an historic district worth seeing.
Over in the Pamplona direction, Olite and Estella have points of historic interest and are picturesque. They are fairly lightly visited, though I believe Estella gets some pilgrimage traffic.