Home Base for 10 days

The regions of Normandy and Britany based in Bayeux and St Malo but rent the car when you get there.
Or the Dordogne area you can get there after renting a car in Bordeaux.

I'm a bit biased but I suggest the Bordeaux area. The city itself is a gem with museums, wonderful food, great markets, wine (of course) and lots of seasonal activities (check out Lost in Bordeaux's website.) From Bordeaux, you could visit the coast OR inland to the Dordogne. On the coast you have gems such as Arcachon (a short drive OR train ride), Biscarosse among others. Further down the coast you have Biarritz and St. Jean de Luz (about a 1.5 to 2hr drive OR you can take the train). To the east, you have St. Émilion, which is an incredibly charming town as well as a renowned wine area (about a 30 min drive). You could then head further in to the 'Dordogne' region which is filled with historical chateaux, battle sites (the last battle for the 100 years' war was in Castillon la Bataille), great market towns (Ste. Foy la Grande on Saturday mornings or Issegeac on Sunday mornings are my favorite.) Sarlat would be the furthest I would go for a day trip if I were staying in Bordeaux however, it would be about a 3 hr, one way drive. You could also take the train from Bordeaux to Sarlat. There are many, many charming and beautiful villages steeped with history on that train route though most you would need a car to access.
Happy planning!

Just came back from a cruise and I visited Rouen and Bordeaux.

I would recommend either place. Both are beautiful and interesting.

I was on a shore excursion and only had a few hours in each city but would not mind going back for more days and to explore further.

A lot of the obvious-to-me options from Bordeaux would be considerably longer than an hour away. When your base isn't Paris or Nice, the 1-hour cap is rather limiting (even if you fudge the 1-hour maximum a bit). I'd definitely recommend Nice for its large number of nearby side trips, but it's much more than 3 hours from CDG.

I'd be inclined to split the ten days across two bases.

One possibility I have come up with is Rennes, a nice city in eastern Brittany. Via train or rental car you'd have access to St-Malo, Dinard, Dinan, Fougeres, Vitre, Mont-St-Michel, Le Mans, Vannes and probably also Nantes. I've been to all those places except MSM and found them worthwhile, though St-Malo was horribly touristy in the summer. One disadvantage of this area is that it is cooler and wetter than average. I haven't researched the weather data for May; you should do that.

I imagine folks who know France better than I do can come up with other potential bases.

We went to the Alsace area and Loire Valley in May, but while we enjoyed those areas, there would not be enough to do in one of those areas for 10 days.

One of our favorite areas of France is Southern France - the French Riviera or Cote d'Azur. If you do not already have your plane tickets you could fly to Nice. The transportation in this area by train or bus is really great. We loved Villefranche sur Mer, Monaco, Eze, and Nice. You could also easily get to Antibes, Cannes, and Menton. We have been to this area twice and would love to return - the scenery is spectacular with the mountains and the sea! The food is wonderful!

Thanks for the ideas everyone! I want to see it all!

The winner seems to be Rouen, as it ticks a lot of the boxes. Could probably spent 2 days in town and then day trips to Giverny, Etretat, Honfleur, some of the beaux villages, etc. Looking forward to it!

Rouen is quite beautiful.

I enjoyed wandering around the coastal towns/resorts of Deauville and Cabourg. Like Etretat, they're likely to be a bit more than an hour away.