day trip from Murren to Bern?

Bern is an enjoyable and feasible day trip from Murren. Your 8 yo may enjoy the Paul Klee modern art museum in Bern. Klee's art is pretty weird to begin with, but the museum also has a section for kids to go nuts with art activities. I also loved Bern's Einstein museum.

Yes, Bern is worth a day trip.
It is easy enough, just descend the mountain to Interlaken the same way you went up, and grab a train to Bern, which will connect with the train you are getting off.
Bern station is at one end of the old town, which is in a U-bend of the river. Just walk out the station and start exploring.

Two things not to miss. The clock tower (photo; https://upload.wikimedia.org/wikipedia/commons/1/17/Zytglogge_01.jpg ). Try to be there when it strikes the hour.

The bear park. This will be a hit with your 8-year-old. "Bern" is mangled old-German for "Bears" (modern German: Bären) and bears have been kept by the city since it was founded. The modern animal-friendly bear park is at the far end of the old town, across the bridge.
Website: https://www.bern.com/en/detail/the-bearpark

Day 7: depends on how early your flight is out of Zürich, and whether you can get from Mürren to Zürich airport in time on your day of departure. You could make this your Bern day, and stay in Bern, which is a lot quicker to Zürich airport.

The Zytglogge is currently covered in scaffolding as it undergoes repairs. It's due to be back in the public eye in June 2018, but you may want to be sure it's there before you go.

You say Days 3-6 will be in Murren. Does that mean only 3 nights? So you will only have 2 full days. Unless the weather is just absolutely horrible I would not waste a day of the beautiful Berner Oberland (Murren, Wengen, Latuerbrunnen) to travel anywhere else. There's so much to see: Almendhubel, walk down to Gimmelwald through all of the farm animals, check out the cheese hut, Trummelbach Falls, Schilthorn, Jungfrau, Wengen, Pfingsteggbahn (longest toboggan run in Europe), Ballenberg, etc.

Spending lengthy journey's on a slow moving panoramic train sounds nice for us older folks, but maybe no so interesting for an 8 year old. Fun train ride with kids try this one: https://www.rhb.ch/fr/leisure-travel-and-excursions/groups-schools/charter-carriages/open-scenic-carriages

Also Geneva is definitely not the Swiss Experience that most tourists have in mind.

If you stay your last night in Zurich then have a look at the Hotel St Josef which is squeeky clean, near the station, includes bfast and is quite affordable for expensive Zurich. From Zurich you can head out to see Rhine Falls in Schaffhausen.

Thanks everyone! I meant 4 nights in Murren, so that gives us three full days there (plus half a day on arrival).

I've jotted all these ideas down. Especially the list for the Berner Oberland - I love the idea of spending all my time hiking, but my husband and son are used to cruise ship "itineraries" and they like to be on the move. So I need to come up with a daily plan in terms of activities. I love the idea of hiking down to Gimmelwald , the cheese hut, etc.

Download the Swiss Public Transport App. Its a lifesaver. Put in your location and where you want to go and it will tell you every step you need to take to get there. What time the train leaves, how long it takes, everything.

Thanks Keim that is exactly what I needed! I've been poring over the RS guide with the estimated transportation times, but something more specific sounds perfect.

That Swiss Transportation System app for your phone is called SBB Mobile.
On your computer, the web site is www.sbb.ch/en

I've been poring over the RS guide with the estimated transportation times

Anything in a guide book can only be to give you a rough idea. ALWAYS ALWAYS check transport times and prices for routes you want to take with the company operating the train etc.

In Switzerland, all rail and bus lines are on the SBB website: www.sbb.ch/en
All rail lines have at least one train per hour, so when you go anywhere you know there will be a train.
As you were asking a question starting from Mürren, I assumed you were already familiar with the train system and how to get to Mürren.

Definitely second all the recommendations here. Bern is an easy day trip, but oh my goodness--you will NOT want to give up a day of the Alpine scenery from the Murren area once you are there! I would add the North Face hike from Allmendhubel and the views from Maanlichen. I'm not sure I'd go up to Jungfrau again, especially since it seems like Schilthorn is adding the kinds of wintry activities (sledding, ice cave, etc) that seemed more unique to Jungfrau (between the two, anyway--Titlis by Lucerne is a similar experience). I also loved Trummelbach Falls, which is a doable walk from Murren (the bus is faster, but the walk through the valley is gorgeous on a sunny day).

We stopped in Bern as a day trip on our way to Mürren. There are plenty of luggage lockers at the train station. Rick has an excellent Bern city walk in his Switzerland guidebook that begins from the train station.

Look up the train travel time on the SBB website: www.sbb.ch/en from Lucerne to Bern and then from Bern to Mürren. The Swiss trains are easy to use and run like clockwork. This way you will not have to travel back and forth from Mürren to Bern and you will not loose one of your amazing Alpine days.

A clarification: if you are using either the Swiss Rail website or the Swiss Rail app, it doesn't just have information on trains, or even just other motorized transit like buses. It has total information on getting from point A to point B anywhere in the country. For instance, if you put in your exact destination in Bern, it will tell you how to get from the Bern train station to that destination, whether by tram, bus, or walking.