Apartments and holiday homes
If you stay for a week or longer, an apartment might be a good alternative to a hotel room. The supermarket in Mürren allows you to get your groceries to do your own cooking. But you'll also find several restaurants in Mürren. Some might be closed in low season though.
The landlord or rental company will provide a free Guest Card upon arrival. The card provides various discounts on local attractions and a few cableways. Note that if you own a rail pass, you will often get a better discount.
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