Yes, I would say Brighton is the perfect comparison to San Sebastian, in size and also that whitewashed, regency feel. It is not a big city as it is geographically restricted by all the mountains. Perhaps Brighton is a touch bigger actually (wikipedia has Brighton at 80 km sq and San Sebastian at 60!)
The route between San Sebastian and Bilbao has some really lovely fishing villages, lots with great beaches. Zarautz is the nearest to SS and has a huge beach with a big promenade, and Lekeitio has a wonderful island in the sea that you can walk to at low tide and then get stranded on as the tide comes in! mutriku (much nearer bilbao) has really lovely salt water public swimming pools on its beach, but is very very sleepy. I am afraid I have no idea how accessible these places all are via public transport though, especially from the airport - the road between the villages is a windy coastal road so taxis charge a lot and the little towns don't have stations.
All the towns on this coast including San Sebastian have a very popular surfing scene which your 9yo might really enjoy watching, it can be quite impressive! the good thing about San Sebastian and Lekeitio's beaches though is that they are protected by islands in the sea which prevent the big scary surfing waves from overpowering everyone in the sea (although San Sebastian has one unprotected beach for surfers).
Does it have to be Spain? Just over the border in France, Saint-Jean-de-Luz is a much smaller town with a lovely square and a really great, quiet beach. Very French though!
I hope this helps, I live here so if you have any questions PM me and I'd be happy to help!!