The London Eye is fun...
Plenty of the London museums are free, so definitely worth visiting, especially if you have a special interest in things like the theatre (I really enjoyed the Theatre Museum, and it only takes an hour or so to see).
The Churchill War Rooms are fascinating if you're interested in history. I went this summer and would definitely recommend.
IF the weather is OK (I agree with the PP about no guarantees at any time of year!) the parks in London are beautiful. As well as the big parks, there are LOADS of tiny ones hidden away in unexpected places, and they are really cool.
Oxford and Cambridge are doable day trips from London (not on the same day!) either by car or by public transport, depending on whether you are hiring a car or not. They're very pretty cities, and pleasant to wander round and explore.
Personally, I wouldn't recommend Stonehenge (sorry PP!)- a huge disappointment, especially when you realise it was "repaired" by the Victorians! But I second Windsor Castle! Well worth a day out, and Windsor itself has some nice shops!
If you can run to the expense, see a show in the West End!
Accommodation wise, pick somewhere near a tube station. When all lines are running, everywhere is accessible with a maximum of one change on the tube network, so it is the best way to get around. Get an oyster card for the cheapest fares.