Bad weather options in Mürren

First, I'll repeat the standard advice (because it's correct). The weather in that area can change rapidly, so be prepared for when it gets good. Ask at your hotel for advice (since they live there, they will understand the weather patterns better than you can). And check webcams frequently. The weather can start bad and become good later in the day, or vice versa; be ready to shift your activities accordingly.

For a day (or part of a day) when it's cloudy or damp, some good things to do include the Trummelbach Falls, a hike on the valley floor (in Rick's book), a day trip to the Ballenberg Open Air Museum, the Lauterbrunnen Talmuseum, and just walking around Muerren and Gimmelwald (using Rick's walks so you know what you're looking at). If you want an excursion that's less expensive (so you don't waste money if the weather turns bad, but you get rewarded if the weather turns good), I recommend the Allmendhubel.

Note that while the Ballenberg museum was fine in light rain, the paths got muddy in heavy rain, so it wasn't much fun once it started coming down more intensely. It was fine before that, as you can go inside the houses and climb the stairs to see upper levels (you can go into any place not actually barred).

Whatever you do, don't waste a lot of time and money going up the Jungfraujoch or the Schilthornbahn if the weather is lousy. Look at the webcams at the bottom before going to the top.

There are some waterfalls in the Lauterbrunnen valley. If and only if you have the Swiss Travel Pass, I would still go to Schilthorn because the James Bond stuff is interesting and its something to do. If you are talking about bad weather in terms of poor visibility, I would still visit Gimmewald, Wengen maybe Kleine Schedegg and enjoy the villages. Hope the weather improves for you.

We are here and the weather is fabulous! Lesson learned. You can’t necessarily go by the weather forecast just play it by sight.

And if you want a "real" weather app try meteoblue...