A night around lake Geneva

Look at a map, and you'll see that there are multiple towns and cities along the lake, including Lausanne, Montreux, and Vevey (from larger to smaller). All have trains direct to Geneva airport, starting early in the morning. Depending on when your flight is, you could stay in any one of them the night(s) before your flight, and still get to the airport easily the next day. Check train schedules at Swiss Rail, to make sure your chosen place will work with your flight schedule: https://www.sbb.ch/en/home.html

How large or small a place do you want? How much do you want to spend on accommodation? Other than relaxing, are your drawn to any sights? For instance, Lausanne has the Olympic Museum and the Museum of Outsider Art, Montreux is very near the Chateau de Chillon, and Vevey now has Chaplins' World, a museum about Charlie Chaplin.

If you want a view from your hotel, be prepared to pay more than if you can "make do" with nice views while walking around, and no view from your room itself (that's what I had in Lausanne). Also note that while Lausanne is a larger city, and you probably won't want to stay where I did in the center of town up on the hill, the area around the water (Ouchy) is quieter, and sounds more like what you're looking for.

Good questions.
Our flight is in the afternoon, so we will have the afternoon and evening and the morning. We are not particularly interested in sights. More of a walk through a park or on a boardwalk and a good coffee on the shore.
I was looking at http://www.eden.ch which is very close to the station and the airport CHF 256.5.

Trains from Montreux directly to the Geneva airport take 90 minutes and leave every 30 minutes or so. Very walkable town.

I will check Montreux. I would appreciate ideas cheaper than mine :).

We stayed in the lakefront town of Morges, closer to Geneva than Montreux, the night before flying home from Geneva airport. A lake view room there was less expensive than in Montreux or other places along the lake, and the lakefront promenade was very pleasant. I would happily stay there again.

Morges is famous as the place where Audrey Hepburn spent thirty years, until her death in 1993 ( she actually lived in a nearby village). There are some Special exhibits this summer in the Morges museums. We may try to make it there ourselves when in Switzerland this summer.

https://www.forbes.com/sites/michaelshiels/2017/04/24/givenchy-and-hepburn-a-match-made-in-switzerland/#1b267fba6317

LOVE the town of Vevey. Picturesque and not as crowded as some of the other town along the shore.

We went to Chateau Chillon in Montreaux and enjoyed it and the views very much. The only parts of Montreaux I saw was between the train station and Chillon. I know people like Lausanne for cheese and it is a wine area. When I go back to Switzerland I plan to go to Lausanne. I was in Geneva and other than the CERN museum, it was my least favorite Swiss town.
You will enjoy your week in Lauterbrunnen. I stayed in the Lauterbrunnen valley for four days and it is pretty amazing.

Try to look up online. Did the same thing and was able to find a good and nice place.

We decided on Montreux. There is no point of being too close to the airport, we will have enough time from checking out to a our flight.
I was surprised that most of the places are already booked a month in advance. I guess you need to do it earlier.