One of the best things you can do in Mürren is to go hiking. It's a hiker's paradise. There are miles of beautiful trails starting right from the center of town. They're all very well maintained and marked, so it's easy to hit the trail. The only challenge is choosing which trails to focus on. You can take overnight trips to nearby huts, long dayhikes, or short easy walks. There's an easy walking trail from Grutschalp to Mürren that should only take a couple of hours. Walk along the Lauterbrunnen valley floor on cloudy or rainy days to see the Trummelbach falls (which you pay to get into) along with dozens of free waterfalls! Spend part of a day walking through Gimmelwald to see typical Swiss farm life. Or there are more rigorous hikes to Rotstock hut, Obersteinberg, Schilthorn, and much more