Book an apartment or home
Most rentals are cheaper than hotels. You either go out for dinner, or cook your own meals. Supermarkets and many restaurants are available in Brig.
You will receive a Guest Card upon arrival, granting discounts on various local attractions and activities. Also be sure to check your rail pass for discounts.
Interhome offers thousands of holiday homes and apartments across Switzerland. They offer discounts on both early and last minute bookings. That makes the Interhome website a good place to start your search:
https://www.interhome.com/switzerland/: all holiday homes
https://www.interhome.com/switzerland/early-booking-discount/: find the discounts for early bookings
https://www.interhome.com/switzerland/last-minute/: find the last minute offers