Welcoming Suggestions Within 4 hours of Interlaken

Could you add more info? Where are you coming from before Murren? I would actually add more days to Murren, so you can actually experience being in the Alps. Lucerne is accessible by train and delightful.
All of Italy will be quite warm in July, unless you're in the mountains.
Pls add some details!

Fair enough. We don’t have flights booked yet, but it will most likely be that we will fly into Zürich and spend our first night either in Zürich or in one of the stops on the intercity route to Interlaken before heading to Murren. 3 nights in sleepy Murren is about all we can ask of a teenager. Thanks

Oh, I disagree , once the teen starts riding cogwheel trains, gondolas, and perhaps even parasailing in Murren! There's even a sports center in Murren that most hotels give you access to.

Can you fly out of Geneva? I think a teen would enjoy the lake towns near there but I could be totally wrong. How many days do you have?
Lucerne felt small and charming but was very lively and puts you within reach of Zurich for outbound. Good shopping. I;d have no qualms about letting a teen that age wander around the town herself, while you sit along the water and enjoy the view.
You may be hard pressed to do all on your list with three nights in Murren.

sleepy Mürren? Have you been there? I see you also use "serenity". I've never heard it described that way.

What does she like?

Climbing on the outside of a cliff? Check!

Parasailing (tandem)? Check!

Trottibikes? Check!

Seeing waterfalls? Check! Going behind them? Check! Climbing inside rock to see them? Check!

Lake boats with lots of interesting people on that boat, the beach, other boats? Check!

Cogwheel Train rides? Check!

Indoor and outdoor sports at the Sports Centre, and lots of kids her age? Check!

Mountain biking? Yup, you guessed it.

Fondue and pizza at the Youth Hostel in Gimmelwald? Just a few minutes walk or bike ride.

The Thrill Walk up near the summit of the Schilthorn?

I haven't finished but I'm sure I wouldn't call it sleepy Mürren. But it is true I don't know your daughter.

We are flexible on the timing of our return, but likely have another 5-7 nights after Murren. As far as length of stay in Murren, it turns out we aren’t the only geniuses wanting to escape the heat in late July - lodging options fitting our parameters are extremely limited.

Steve, lodging options in many European areas are very limited this summer due to the pent- up demand. I'd suggest drafting an itinerary, booking lodging, then modify. Many folks here on the Forum report what you're finding. Hope it works out!
Ps
I'm an OSU alum and OU alum!

3 nights in sleepy Murren is about all we can ask of a teenager. Thanks

One of the toughest posts to answer is when someone says they are traveling with a teenager. Only you as parents can answer the entertainment question of what they like. I would challenge your daughter to research to find something to do in the areas you are traveling. As Nigel wrote there is a lot to do in the BO. If all else fails, there is always the Bahnhofstrasse in Zurich, but be sure your credit card limit isn't too low :)

If it were me, and flying into Zurich, I would go directly from the Zurich airport rail station to Luzern ( just over an hour by train) and stay there a couple of nights to get the urban vibe (its a small city) and then not leave the Murren area. Or end up in Luzern and take the train from there to the Zurich airport for departure.

Bern is also reachable from Murren.

Have you looked at Lausanne or Vevey? They are close to Geneva and you can easily stay there until the day of your outbound flight from Geneva if that is an option. Lots to do including trains up to the mountain areas, on a fabulous lake to cruise, good restaurants. The lake keeps it relatively cool there.

As mentioned above, Lucerne is also a good choice with nearby peaks and a lake.

I'd choose Luzern just because it fits with your itinerary - easy access to Zurich airport, great scenery and a bustling small city vibe. Just keep in mind that Switzerland's cities are all relatively small - you classified Bologna and Verona as "smaller" cities, but they are comparable in population to the biggest cities in Switzerland (Zurich similar to Bologna, while Geneva or Basel a bit smaller than Verona metro area). Luzern only has about 82,000 and is still 7th largest in Switzerland...

Anyway, there are quite a lot of students in Luzern, so it does have a younger, more urban feeling and there are good restaurants/shopping/etc. Easy to do a day trip to Zurich and/or Basel as well for more of a bigger city feeling.

And side note to address some of the comments about difficulty finding accommodation: yes, there is pent-up travel demand, but prices have also gone up in many popular destinations due to inflation and a big increase in energy/running costs. As a result, there is stiff competition for the more reasonably priced places and booking early is a must. Short-hop flights and long-distance trains are also more full, especially on weekends. We have noticed that many night trains are sold out far in advance for weekends through the entire summer.