Sounds like a fantastic trip. You have seven nights before your travel day to Chamonix to start your TMB. I suggest you spend 4-5 in Mürren (a lovely village with excellent hikes right from your door) and 2-3 nights in a small village right on your way to Chamonix, to give you an opportunity to see and hike in a different area, and also to break up your journey between Mürren and Chamonix.
Two places come to mind. Both offer beautiful hiking, some of my favorites anywhere. We always include stops in both villages when we go to Switzerland to hike, but you will have to choose just one
- Kandersteg is a small town in a valley right on the train line going to Brig, where you need to go to pick up the train to Chamonix. It used to be the main train line between Bern and Brig, but with the new Lötschberg Basistunnel, the main line runs to Visp and then to Brig. I think the scenery is much better on the Kandersteg route---there is still a tunnel, but it is not as long, and you get great views coming in to Kandersteg and again after the tunnel.
The attraction at Kandersteg is the beautiful lake above town, named Oeschinensee. You get there by gondola from Kandersteg ( we walked up once and I thought my family was going to kill me!). There is a beautiful if rustic mountain inn up there where we stay--Hotel Oeschinensee. My favorite hike goes up the near-vertical cliffs surrounding the lake to a hut, the Fründenhütte. The path follows ledges up the cliff, wide enough to feel safe and with cables to hold where you would want them. The hut itself sits near the terminus of a glacier (or maybe not so near these days) and is a starting point for climbers and serious mountaineers. But the views at the hut are enough for us. They sell refreshments and you can overnight there, but we never have done that. Flip through the photos on this site:
http://www.oeschinensee.ch/huttenwandern/
I will not do this hike on a rainy day as the rock may be slippery. So it is not guaranteed that you can do it. If not, there is an excellent, and more strenuous hike going to a pass and hut around the other side of the lake. It is called Blumlisalphütte. This path is a mix of forest, rocks, and scree but not exposed and not bad in the rain. You do get nice views of the lake as you ascend, just not as dramatic as the straight-down views from the other path.
- Bettmeralp is a small car-free village perched on a bench above a valley, much like Wengen. You reach it by cablecar up from the train stop of Bettem, about 12 minutes east of Brig. The attraction here is the Aletschgletscher, the longest glacier in Europe, which is just over a ridge that rises behind the town. You cannot see if from town ( but on a clear day you can see the Matterhorn). There is a nice loop hike you can do right from town: ascend the ridge behind town to a saddle ( from which you see the glacier), drop down a bit and then you do a long traverse above and parallel to the glacier. After several miles you come to a point where you can cross rocks and get right up to the side of the glacier, but don't get too close as ice chunks fall occasionally. From here the trail continues to a hut on a small lake,the Marjelensee. We stop there for lunch or beverages. From this point you can either take a 1-km tunnel under the ridge, or hike over the ridge, to reach the south side of the ridge at a place called Kuhboden. If there is time and the inclination, you can ride a cablecar up to the top of the Eggishorn for fantastic views north all the way to the Jungfrau. Or you can simply continue on a near-level path which becomes a farm track ( muddy) to return to Bettmeralp.
If you have another day here there are other hikes nearby, such as up the Bettmerhorn, or to the top of the ridge at Moosefluh and down to the next village of Riederalp.
Google "Bettmeralp" and Aletschgletscher" for photos.