Sandi,
I recently stayed at the agriturismo Niccolai, just outside the San Gimignano city walls and absolutely loved it. We stayed in an apartment there, I believe in Torre Palagetto. It had 2 bedrooms, 2 baths, a kitchen and a dining room. I can't remember if our kids' bedroom was 2 twin beds pushed together or if it was actually a double bed. I didn't look to see if they have any three bedroom apartments. The agriturismo had a pool. The setting is beautiful, overlooking acres and acres of vineyards. Best of all, it's about a 10 minute walk into San Gimignano itself. From that location it's easy to drive to Siena, Volterra and the Chianti region, among other areas. I thought the price was very reasonable 900 euro for a week in late May. Our children are young (6 and 3), so we didn't have to pay for them-- I don't know if the teenagers would increase the price in any way.
Have a great trip. Tuscany is beautiful and relaxing no matter where you end up.
Ciao,
Meredith