My husband, daughter ( 19 years old) and I will be travelling by car from Germany to Berner Oberland and spending 4 days there in Murren. We plan to hike all four days and take the lifts and cable cars that do not appear to be covered by the Swiss pass. In addition, we will be spending 3 nights in Zermatt and hiking there as well. I do not know what hikes we will be taking but it appears as though we will also encounter lifts that are not covered by the Swiss pass. After leaving Zermatt, we will be spending 2 nights in Lausanne before departing the next morning from Geneva back home to United States. One may ask why we are renting a car but it is very convenient while traveling in Austria and Germany and gives us a considerable amount of flexibility. The question I need advice on is whether to buy a Swiss Pass for a discount on the cable fares while hiking or if the half price card is a better option.
Most people who travel to the car-free villages like Mürren and Zermatt with a car, but want a pass for riding lifts, choose the Half Fare Card.
I can understand why you might want a car for Germany and Austria, but have you looked at the actual cost of keeping it for the ten days in Switzerland? There is a large drop fee for returning it to a different country, plus the Swiss vignette, plus parking for the days you will not be using it at Murren and Zermatt, in addition to the extra ten days of rental.
When my son and his wife joined us in Switzerland after then toured Germany, they turned in their rented car at the last town in Germany (with no drop fee) ) and continued by train. Something to consider.
The Swiss Half Price pass costs 120chf seems to be the one that you would gain some kind of savings.
However you need to research add up the total costs of cable cars you plan to take at full versus half price to see the difference.
For four days in the Berner Oberland, I would buy the Berner Oberland Regional Pass. It will cover all of your lifts and trains area. We hike extensively when there and it always pays off financially for us. There is a supplement to be paid for the Jungfraujoch and Schilthorn. The HFC is good, but I find the BO Reg Pass is a better value when you stay in the Berner Oberland for several days.
You might benefit from having both, though, since you are also going to Zermatt. We used an HFC for the discounts when we went to Zermatt, then also benefited from the extra discount you get on the BO Regional Pass when you have the HFC.
"take the lifts and cable cars that do not appear to be covered by the Swiss pass."
Correction: the lifts and cable cars as high as Mürren on one side (including Gimmelwald) and Wengen on the other side are fully covered by the Swiss Pass. The lifts above that are 50% covered, except for the trains above Wengen to Kleine Scheidegg and Jungfraujoch, which are 25% off.
For Zermatt, looking at the 2017 coverage map, it seems that the lifts are 50% off with a Swiss Pass: https://www.sbb.ch/content/dam/sbb/de/pdf/freizeit-ferien/ferien-kurztrips-schweiz/internationale-gaeste/sts-geltungsbereich_en.pdf
So, the big difference between a Swiss Pass and a Half Fare Card for your itinerary is that with the former, the Jungfraujoch will be only 25% off instead of 50% off, but as a trade-off, all the transport within and between cities, and to the lower elevations in the Berner Oberland, will be totally covered. The other differences are that museums in Lausanne will be covered by the Swiss Pass but not the Half Fare Card, and that with a Swiss Pass you only have to buy tickets for items not fully covered (a nice convenience), whereas with the Half Fare Card you have to buy tickets every time (not hard, as all machines have a half fare option).
I'm not familiar with the Berner Oberland Pass's coverage. As always, you have to do the math yourself to see which is the best deal. If you're unsure, remember that you can't lose with a Half Fare Card as long as your total travel comes to more than 240 CHF.
If you do not get a Swiss Pass, but are staying in a hotel in Lausanne, you should be getting a pass that is good for local transit in Lausanne. It will not cover transit to Montreux or Chateau de Chillon.