Welcome to MySwissAlps Laura88keys!
Great that you have the opportunity to visit several Swiss areas, especially given your family history :-). I think your plan to stay in one base town for a couple of nights, making day trips from there, and then moving on to a new region, makes perfect sense. It's more relaxed than switching hotels nearly each night.
The timetable is a great tool to plan journeys. You can use it to find out how long day trip from several towns take. That can help you decide what is and isn't doable for the three of you. This page explains how to use it: https://www.myswissalps.com/timetable.
Lausanne is a good choice, with easy connections to both Geneva as well as the eastern part of Lake Geneva. Please find information and tips below:
I'd choose either Bern or Thun and not both, as they're close to one another. Bern is larger and offers good connections several regions, including the French-speaking area of Neuchatel and its lakes. Thun is closer to the Jungfrau region, plus it's a good base to explore the western areas of the Bernese Oberland. Whether or not a base within the Jungfrau region is recommended, really depends on the trips and sightseeing you have in mind. As for Charmey: a single trip Charmey-Bern takes 1h40 to 2h15, and you can leave as late as 21:40 pm. You can find more information about the cities and regions below:
Lucerne and its lake are wonderful and have lots to offer: https://www.myswissalps.com/lucerne.
Basel is pretty close to Aarau. That could be a reason for you to skip Basel as a base town, since you're set on staying in Aarau. Basel has a beautiful old town though, which would be worth day trip from Aarau: https://www.myswissalps.com/basel.
As for north-eastern Switzerland: Zurich (https://www.myswissalps.com/zurich), St. Gallen and Glarus would all be good. You can spend a few days in each of them. If you prefer to strongly limit the amount of base towns, you could opt for St. Gallen only, as it's sort of in between Zurich and Glarus. It's a beautiful town, famous for its monastery and library. Zurich is 1h to 1h15 away by train, Glarus 1h10.
A nice order could be: Lausanne - Bern / Thun / Jungfrau region - Lucerne (I suggest traveling via Interlaken and the Brünig Pass: https://www.myswissalps.com/goldenpass/gettingthere) - Aarau or Basel - Zurich/St. Gallen/Glarus. Of course there are many more ways to compose your itinerary; this is just one.
These practical planning tips may help you out: www.myswissalps.com/plantrip/doityourself.
More information about the Eurail Global Pass is here: https://www.myswissalps.com/eurailglobalpass.