I was just in London at the beginning of the year for a few days, and we stayed at a really cheap and friendly B & B at a terrific location in Greenwich- but that might not be central enough for you. It’s right on the park. Strictly no frills, but they do have an enclosed carpark:
Greenwich Parkhouse Hotel
If you’re looking for a proper hotel, the Holiday Inn Express across from the Dome occasionally has some brilliant deals on, although be aware that it is in the middle of absolutely nowhere and you really need a car in order to stay there:
Holiday Inn Express, Greenwich
PS: The Chinese restaurant underneath the hotel is FANTASTIC. We were the only non-Chinese table in there when we went.
And just for laughs, there’s an Ibis in Greenwich too: http://www.gmt2000.co.uk/meridian/sleep/slphia1.htm
Hmm…The tariff on that last link seems a lot lower than what I remember being posted outside the hotel as I lurched drunkenly past it on my way home a few weeks ago. If you can get that price, it’s a brilliant deal.
I’m sorry those links are so SE-centric, but all it’s very true about London being a series of linked villages, and all I know of it is the sahff east bit.
Anyway, because my mother had never been to London, what we did was take the afore-mentioned boat trip from Greenwich to the London Eye and not only did she really enjoy it (and my mother is, embarrassingly, an afficionada of the video recorder of near-Niponic proportions), but I myself was pleasantly surprised. They give you a good overview of the sites and it's surprisingly affordable (we paid 5 pounds something one-way, but the return is not much over a pound more), and then, of course once you get there, you're central to a lot of the tourist sites anyway.
We went from there to the [Wallace Collection](http://www.the-wallace-collection.org.uk), which I thought was brilliant but the others didn't enjoy so much (You have to be really fond of painting- I'll admit some of it was a bit Rococco for my tastes, but the 16th century collection was FABULOUS. I hear the cafe's nice too, but it looked a little black turtleneck-improvisational-jazz-fan to me). Later during the trip, we also took in the [Imperial War Museum](http://www.iwm.org.uk), which I can't recommend vociferously enough- although I think my mother's German boyfriend didn't enjoy it quite so much; there is a poignant and very harrowing Holocaust exhibit and he doesn't like to mention the war.
Hope this helps. We had a great time. Saw more of the inside of various pubs than the sites, though.
Happy hols.