Recommendation City between Chamonix and Murren

You do have a lot of time in the mountains so if you are looking for something a little different to break up the trip you may want to consider the town of Sion. It's in Valais and is about halfway between Chamonix and Murren on the rail line.
Sion is a lovely town in the middle of the vineyards of Valais. The vineyards in this region are very old and date to the time of the Romans. They are terraced up the hillsides in this region and they are just stunningly beautiful! Sion has a very pretty old town core that is very walkable. There are some tourists but it's mostly local people. In the middle of town are two hills; one topped with a ruined castle that has some intact rooms with very interesting paintings, and the other hill is topped with an 1,000 year-old church, the Velere Basilica, that has been altered very little since it was built. The organ is from the 1500s and still plays...if you're lucky you'll be able to hear it played - it's the oldest working organ in Europe.
The surrounding area has many castles and ruined towers that can be toured and explored.
Mont Fort is just adjacent to Sion...it's the highest point in this part of Switzerland and from the top you can see into Italy and France. Have a drink at the highest bar in Europe!
If you hit it just right, you can go to the Alphorn Festival in Nendaz just above Sion - 22 - 24 July this year. Several hundred Alphorns playing in the meadows above the valley...food, handcrafts, few tourists. I've never experienced anything like it before...one of those unique experiences you can have when traveling!
That said, Kandersteg is beautiful...personally I can never get too much of the mountains of Europe!

You could try to ship luggage from Aosta to Chamonix and take the cable car from Courmeyeur to Chamonix (weather permitting).

The train from Chamonix to Sion is spectacular, and Sion is well worth a day.

4 nights Murren + 5 nights Wengen is a long time for one valley (certainly a great one). Consider splitting that time with Zermatt. Go from Sion to Zermatt, then Zermatt to Murren.

Consider staying in Interlaken one day to visit the lakes and get a little break between mountain towns (like Sion after Chamonix).

You could also go from Zermatt to Murren via Andermatt (part of the Glacier Express line) and try the stage coach ride: http://www.gotthardpost.ch/index.php/en

If you can make the Alpenhorn Festival in Nendaz it is worth it. Parades, Alphorns, yodelers, dancing, drinking, and eating local foods! It is really unique and worth visiting. Also, good reasonably-priced accommodations in Nendaz.

In my opinion one of only a couple of reasons to be in Switzerland during Juli. Also to be seen in Juli: Blue Balls Music Festival in Luzern, Montreux Jazz Festival, and their version of 'Independence Day' on July 31 everywhere. I wasn't too impressed with Montreux but I have only been there once so I would hate to dismiss it and their little jazz festival is nothing to sneeze at ;)

I wanted to thank all three of you for your opinions. Here is what we have changed so far:

We found a one-week beautiful rental in Villars (a place I had never heard of). This location allows us to spend one full day going to Sion and that area. It also allows us to spend two days going to Lake Geneva to see Montreux (I think the jazz festival), Vevey, Lausanne and another day to maybe do the Chocolate train. The entire time we will be checking the weather reports for Zermatt, and if it is clear, we can spend the day around the Matterhorn. We are still filling in the other days but we do want to spend some days simply bike riding, hiking and relaxing.

Then we head to Wengen on July 15th for five nights and Murren for three nights (I rearranged the two stays and cut it to 8. I know some may think this is a lot of time up there, but we fell in love with the place back in 2011 when we were there for four days).

So this new schedule only leaves three days at the tail end of Murren. We are thinking Interlaken so we can see the lakes, steamboats, Thun, Bern and maybe even Murten. The other alternatives might be Luzern (but we did this in 2011) or Zurich.........or maybe try to stretch it and pass Zurich to stay around Obersee Bodensee and see cities such as Konstanz, Meersburg, Lindau and that nice waterfall. My preference (and the easiest) would be to stay somewhere around Interlaken so we can spend some time down and around those beautiful lakes.

Any other thoughts would be great.

Oh yeah, and Roberts idea about the gondola from Italy to Chamonix is a good idea. We will still have a car at that time so I might surprise my family and drop them off at the lift on the Italian side and then I will take the hit and drive through the tunnel and eventually meet up with them at our hotel in Chamonix. I noticed a new gondola system that has recently opened on the Italian side.

https://www.youtube.com/watch?v=6gMJmh1Icvs

I will have to do this when we are in Aosta, but I wonder if this is separate then the gondola that goes over the mountain to Chamonix?

I am not sure about that festival in Nendaz since the days seem to coincide with Murren and maybe Interlaken (If we stay there at the end of our trip). The train map says 2 hours from Interlaken so we will consider it because it sounds awesome and is the exact type of events we are looking to find while in that part of Switzerland.

I will have to do this when we are in Aosta, but I wonder if this is separate then the gondola that goes over the mountain to Chamonix?

Yes, that's the same one.

The Trient Gorge, at the Vernayez train stop from Chamonix outside Martigny, is worth a quick look if in the area too.

Now that I know you're staying in Villars (I've been up there...very pretty place but it's a bit of a drive down the mountain!) I would suggest a couple of other places to go in the area. I'm assuming you'll have a car??
Definitely take a drive up to the St. Bernard Pass where Hannibal crossed the Alps and the Romans marched over into Europe. It's pretty stunning and the famous St. Bernard monastery is there with the original breed of dogs! You can visit them and take them for a walk in the mountains if you wish! Very fun place.
Verbier is a great, if not crazy expensive town, and you can drive up the road through the ski slope to a restaurant called Croix de Coeur. Stunning views across the Swiss, French, and Italian Alps.