In Visp there is a small and good value hotel called “Visperhof”. They provide free WLAN access in the hotel and free parking in a garage plus a superb breakfast and everything for only 65,-€. Good deal.
Visp is located very central in the western Alps at the spot where Mattervalley and Saasvalley meet the Rhone valley. All higher mountains of the western alps can be reached very fast from there within under 2hrs. This is a good and central base to start from before and come back after some days climbing different mountains.
They are also very flexible with the booking and you don't have to stay all nights in a row, but can leave your room when your out for mountaineering and come back another day.
Possiblities:
You can get to the western part of Bernese Alps and reach Kandersteg in 1:31hrs and climb Doldenhorn or Blümlisalp.
You can access the southern side of Bernese Alps and get to Blatten and Fafleralp in 1:30hrs to climb Bietschhorn.
You can use the Simplonpass to get into Italy and reach Macugnaga in 1:57hrs. and take a look at Europes highest iceface, the 2,7km high Monte Rosa eastface (or even climb it?).
Zermatt is reached in 1:16hrs. and Saas Fee in 0:59hrs. From both villages you are in touch and have acess to a big amount of 4000m mountains located in the following moutainranges: Weissmies-range, Mischabel-range, Monterosa-range, Weisshorn-range.
To get into France and the city of Chamonix is possible in 1:53 hrs. Montblanc-range offers everything.
To get into Italy and the city of Aosta is possible in 1:59 via the S.Bernard tunnel.
If you are looking for something directly in Saas Fee or Zermatt with comparable quality, you'd have to pay more. I wouldn't recommend a stay in the very touristic spots because not only accomodation but also food is way more expensive there than in some spots not far away.
From the saved money you can easily rent a car and drive around here and there.