Once you decide how many days you will spend in Bilbao it will become easier to pick the right area to stay on your visit.
For shorter trips like 1 day and overnight in Bilbao there are many good hotels in the city center. This area will be a good option because you can just walk to the main Bilbao sights. My pick of (not cheap but great) hotels would be the boutique Hotel Miro, Melia Bilbao, and Hespria Bilbao across the Guggenheim.
For conveniently located and good value hotels in Bilbao you should look at Jardines de Albia, Barcelo Bilbao Nervion, or Hotel Conde Duque. If you are driving, all these hotels I mentioned have good parking facilities.
For a longer trip you could get a real feel of Bizkaia by staying outside Bilbao city. Palacio Urgoiti is a small rural (upmarket and so not cheap) palace close to the airport. But many find it a surprisingly good value room rate for its ambience and facilities that include a golf course.
Also look at this site if you are the type who likes the thought of staying in rural Basque accommodation that is basic, but authentic and comfortable, without breaking your bank.