Children under 11 travel free on London Transport, something I didn't know before I went last time with ds (aged 7).
Transport in London is pretty pricey and the Tube map bears little relation to where the stations are (relative to each other I mean) above ground, so don't be fooled into thinking you need to get the Tube. Best thing to do is to base yourself in one area, explore it on foot and then maybe hit a few highlights elsewhere.
We stayed by Tower Bridge, and there's a lovely area just beside it called St Katharine's Dock, which isn't too touristy. You've got the Tower of London (not cheap) right on your doorstep and then you can walk down to the Thames to see the Globe, the Millennium Bridge, St Pauls - if you're still going you can then walk to Pudding Lane (this was v important to my ds as they'd just done the Great Fire at school).
Alternatively you could stay near the Natural History Museum - you've got some fab places to go there inc the Science Museum and the V&A. This is probably where we will go next time (as my DH is in a wheelchair, I'm keen to stay in just one area of London rather than tackle public transport).
VodIsGod - those apartments look great. They are definitely on the list for whenever ds and I go just the two of us again!
We've stayed in the Novotel at Greenwich before and you can get reasonable deals on family rooms.
Things to do for free - if you've never been to London, you'll probably enjoy just walking the area that includes Leicester Square, Trafalgar Square (National and Portrait Galleries here too) down to Number Ten and then over Saint James' Park to Buckingham Palace then along Piccadilly. It's a long walk but you'll see loads and you can stop for a rest/coffee etc along the way.
I really like Sir John Soanes House and the Hunterian Museum for quirky museums that don't exhaust the children in the way that, say, the British Museum can do.
If you don't mind a train ride stay at Travelodge at caterham whyteleafe which is literally next to the train station (but no noise) and train goes directly to London Victoria. If you wanted to drive there is good parking at.the hotel.
We always stay there when we go to London. Occasionally we pay only £29 for the room and it's very nice for a Travelodge and in a reasonable area of London.
They have an offer in January for £178 for a one bed apartment (double bed plus £40 for extra z-bed) for two nights.
I have stayed at the Empire Square apartments a couple of times, really nice and clean Borough tube 200 yards away. Borough market, London bridge, St Paul's all nearby.
I think they are cheaper than many hotels and much nicer than basic travellodge/premier inn. They have a kitchenette plus dining table and a Tesco Metro next door so you can save money by eating breakfast/some meals in. And there is a separate proper lounge from the bedroom. If you have to have an extra bed then the bedrooms are spacious enough to fit it in.