Hi Lynn, what a wonderful opportunity, and I hear you on the self-consciousness about the privilege, but you know, we all make sacrifices and suffer somewhere, so I hope you can embrace this wonderful experience and celebrate it without embarrassment
We are a family of 5 and try to travel for 4+ weeks at a time, so I understand the budget. First, I would only look at long-term rentals, which I imagine you already are. And you may be looking out of central areas, but remember, public transportation is so easy and so available, that you can get anywhere from anywhere fairly easily. Alternately, you could look into the price of leasing a van for that time, and use that for the bigger trips, but keep it parked and use public transport for getting in and out of cities. The benefit you have of time means that there is less urgency to stay in the middle of things and hit the ground literally running. You can stay 15-20 minutes outside of a main area (or more) and find much cheaper lodging, and not worry about having to see everything in 3 days (so the "commute" won't really matter - also with the longer stays you can get local transportation passes that will make public transport cost less, and it will be free for the 8 yr old)
One of my favorite cities for livability when traveling is Amsterdam. Its relatively small, very easy to navigate/orient yourself, easy to get around on bikes, easy to get to other places, and fun to just be out and about people watching at Vondelpark or along the canals. Lodging is a killer in terms of price, but you could easily stay north of the city (or west or east) for much less and still have easy access. There are bike trails crisscrossing the entire country. You could actually ride from town to town on nice, paved paths. There's a lake about 30 minute bike ride east of the city along the Amstel river that I rode to with my kids. They swam, played, and then we rode back and went to NEMO science museum and hubby and I had a beer on the roof while the kids were in the museum on their own. That was a great day. We took the train Utrecht and spent a day there. etc etc. My family os from Holland, and when I was 13 and visiting, I decided to ride my bike from the town we were staying in to my Oma's house, without a map and without telling anyone (true, I have a good sense of direction usually and an intrepid spirit). Point is, I could follow paved paths and use signposts for the entire ride (of 3 hours). As an illustration of how easy it is to get around, even for kids
I would also consider Florence or surroundings. Its also a medium sized city but with easy access to other places. Pisa, Lucca, the beach, the Tuscany region, even Rome (for a long day, if you departed early) are right there. I had the same feeling there; easy to become familiar with the city quickly and to develop favorite places to eat and to be able to get around without studying a map.
Southern Spain, as someone mentioned would be good for Spring. You could base along the coast, and then take day trips (in the van ;) or by train) north, west, east.
We have been to Greece, but for us, the beach gets boring fast, so that wouldn't be a draw for us. If I were to go I might choose Crete. You could base in one place and still see the whole island in a month, and there are mountains that get snow in the winter, along with the coast, and villages, and interesting museums in Heraklion...
Of course I love France, but for a base, well, that would be hard. We have been considering basing in Annency or nearby for a longer stretch. Its on a lake, and near mountains, near Geneva (for a dose of city) and other locales in Switzerland, and Tufts has a campus on the lake in Talloires, which I would probably look into for a class for myself if our family did a longer stay there....
Thats all Ive got ;)