Rain and trains...

Does anyone have any suggestions on rainy weather activities?

Hike higher to where it might be snowing? ;-)

jolynna, I am not quite sure what you mean by a "global rail pass". If it is a Swiss rail pass, the answer as to where it is valid is this map: https://www.swisspasses.com/railpass/overviewmap_en.pdf
As you can see, Swiss passes are valid up to Mürren, above that you get a 25% discount. You do not need to reserve seats, Swiss railways don't work that way. Just buy a ticket on the day and show your pass to get a discount.
If you have some sort of multi-country pass, aka "Eurail pass", you will need to check the small print of that pass. But AFAIK most aren't valid past Interlaken, so you will have to buy a new ticket at Interlaken to get to Mürren.

And yes we are having vary varied weather, oscillating between April and June weather. Beginning of this week was wet and cold, about 15°C, weather reports said "snow down to 1000 metres". Today hot sunshine and well over 25°.

Thanks Chris. Yes, it is a Eurail multi-country pass. So it is valid until we get to Interlaken and then I will need to buy a separate train ticket to Murren? Is that correct? It sounds like I don't need a reservation, which is good. Regarding the weather, I'm just hoping for at least a day or even a few hours of decent weather to hike. It's lot looking good though.

The Eurail Global pass covers trains to Interlaken and then offers 25% discounts on tickets in the valley. Just show it when you buy the ticket. It also covers lake boats (on the same days that you count for train travel, if it's a flexi style of pass). On your Eurail map, see the blow-up section for Switzerland, where dotted lines indicate a discount. Reservations are not require for any trains except the Glacier Express. I believe that Eurail's mobile app will also list various discounts, similar to the list of bonuses on the pass purchase page.

Although it's an open-air museum, Rick thinks of www.ballenberg.ch as a rainy-day option.

Thank you Laura. I will check out that museum link.

Rainy day activities: in Lauterbrunnen there is a small museum. Go south from the train station until the road forks, go left, cross the river and it's on your right.

You can also walk up and behind Staubach Falls (the one visible from Lauterbrunnen). Most of the time you will be out of the rain in tunnels and cutouts.
Trummelbach Falls are perfect on a rainy day because you are INSIDE the mountain looking at the falls. Google image it. You can take the Postbus from Lauterbrunnen or walk about 45 min south of Lauterbrunnen. When you see the tour buses, you're there.

Thanks! These are good tips. :) I will check out the falls for sure!