Viamichelin.com suggests two possible routes that diverge after you pass Birmingham. Depending on whether you take the easterly or westerly route, you will be driving very near Coventry and Bletchley Park (WWII codebreakers, but this site really deserves a lot of time) or Stratford-upon-Avon (which I haven't seen but I gather is very touristy), the Cotswolds and Oxford. Those are just a few possibilities. A car is very nice to have in the Cotswolds, because it opens up the villages that are difficult to reach by public transportation. That area could consume all the extra time you have.
I can't speak to what things will be like in December. If you happen to hit miserable weather, Oxford might be a good possibility. There are at least two very interesting museums there, the Ashmolean and the Pitt-Rivers.
There are many, many other possibilities. If you're traveling with children, it would be useful to know their ages.