Mom & 8 year old son trip this summer - Paris and...???

I would stick with France, as the country is wonderful, and you'll be in Paris already. I'm sure you'll get some more responses, but in the meantime, here are a few threads with information on planning a France itinerary with lots of good information.

https://community.ricksteves.com/travel-forum/france/france-itinerary-3b05a264-fbdc-45e9-b21f-33ec31bb17b9
https://community.ricksteves.com/travel-forum/france/france-itinerary-c2e33a78-2868-4d57-a048-ce1bdd0951b1
https://community.ricksteves.com/travel-forum/france/france-itinerary-6abae7f1-03e3-401a-b39e-417ee2d8ef10 (with 13 y/o daughter)

Edit to add: If it were me, I'd go with Brittany and the Dordogne (maybe Bordeaux as well). But I've been to Provence, the Loire Valley and the Alsace region before. Of course, there are many other areas in France that are worth seeing. You also might want to look at this link, and scroll down to "At a Glance," which will give you a quick summary of the various regions. https://www.ricksteves.com/europe/france

Erin E,
What fun for you two! Well, IF you wanted to rent a car I would say the Dordogne would keep you two busy for a week, with lots of things that an 8 year old would love, but a car is a must there. So for someplace else.....

For another location, I think Munich would do for 5-7 days. The Deutsches Museum there is full of really cool science and technology stuff. It is one of the best I have ever visited. My kids loved it and spent hours there. The Nymphenburg Palace and the Treasury at the Residenz the are very impressive, especially all of the Treasury collections. The glockenspiel at Marienplatz, the Viktualennmarket, the parks and swimming areas, the beer halls (noisy, fun, iconic places to eat...not all are full of drunken tourists). I would go to Rothenburg as well, for one night or a long-ish day trip. You can walk along the top of the walls, take in one of the terrific puppet shows, and visit the St Jakobskirche to see the carved wooden altarpiece...magnificent. And just a wander around the medieval streets is impressive. Check out a guidebook about Munich (and environs) to see if you two could have a good time there. I know I did!
Amusez-vous bien!

Take a look at Aarhus. For Scandinavia, it's less expensive. A quick search on AirBnB shows a range of $600-$800 for a one-week stay. They have danishes there, great modern art and archeology museums (including a bog body!), varied architecture (DOKK-1 is a very cool waterfront government building worth checking out--no pun intended, there's a library there), great public transportation and everyone speaks English. Venturing further north to Skagen, you could go and see where two seas meet but don't mix. Lots of other outdoor places to visit closer to town, including the Infinite Bridge.
From Paris, it looks like the easiest thing is to fly into Billund, where Legoland is, then take the train to Aarhus. Or you could take the train through Hamburg. Sadly, the train from Germany to Denmark no longer goes on the ferry (that was cool!).
Here's the tourism site:
https://www.visitaarhus.com

Another suggestion if you haven't already been: San Sebastián. Lots of interest to kids (aquarium, amusement park on the hill, beach right in town), plus great food (tapas crawl through the old town!), affordable if you stay away from the center. We were in an apartment up on the hill near the Intxaurrondo metro station. We walked into town the first morning and took the bus after that.
You can day trip to Bilbao to see the Guggenheim museum there, as well as the Vizcaya bridge.

Another city my son and I really enjoyed was Turin. Again, we stayed in an apartment outside the city center (https://www.airbnb.com/rooms/7251319?source_impression_id=p3_1737831279_P3n6wl1Daq-WepMj) and got around easily on foot and by public transportation to the cinema museum (rooftop viewing platform!), car museum and what I believe to be the original Eataly. We skipped the old Fiat factory with the rooftop test track. To travel between Turin and Paris, you could take the Bernina Express with a one-night stay in Zurich (which is expensive, of course, but one night won't blow the budget, and you'll get to see some mountains) and 1-2 nights in Milan (because they have an awesome science museum). Here is the info on how to get to Turin from Paris:
https://www.seat61.com/international-trains/trains-from-Paris.htm#Paris-Milan

So many great suggestions, thank you all!

We have already been to San Sebastián and Turin, and really enjoyed both cities.

Aarhus sounds incredible! We went to Stockholm a few years ago for 5 days and wished we’d had more time to explore more of Sweden. I’m not sure I’m keen to get on another flight after we land in Paris, but I’ll add it to my list for a future trip!

Munich would be a good stop; I haven’t been in a long time and my son has never been to Germany. I am leaning towards staying in France, just because there is so much to explore outside of Paris. If we don’t go to Chamonix, maybe Provence? I have the RS France book and also looked at Brittany or the Loire, but it seems like for those regions, a car is necessary.

I took my 11 year olds each to Europe individually - 2 boys and a girl. From Paris, each time we traveled to Lauterbrunnen, Switzerland. One of the places we stayed there was a private room at the hostel. The other hotels were not overly expensive in the May to June timeframe which is also when we traveled. There are wonderful mountain trains and lifts and great hiking - long and short, plus the Ballenberg open air museum and a boat ride on the lake covered by the rail pass, so it is very much about being outside. At the time I was there, I was able to buy a rail pass and then get a family card that allowed a child under 12 to ride for free with me. We finished our trips in Munich - traveled by train - and all of the kids loved that too.

Brittany would be interesting. Perhaps head for Mont St. Michel then down to St. Malo for a few nights -- cool beach with offshore islands you can walk to and a big natural swimming pool on the beach during low tides. and as an old walled town interesting. From there you can take a bus or boat to Dinan and Dinard and walk to St. Servan. You could do a bus trip to Mont St Michel from there. when we did this by train from Paris, we spent 4 nights in St. Malo and then traveled by train to the Carnac area so we could tour the neolithic sites.

The loire would be interesting with its many varied chateaux; you can join tours to visit the chateaux.