I think it generally costs about £1 for 20 mins parking in central London (it varies by borough). That's expensive! Parking restrictions are in place on Saturdays until early evening in some areas so it will only be free after 6.30 pm or so. Also, bear in mind that certain areas in the City of Westminster only allow payment by text messenging (I haven't really got a clue how this works). Parking on Sundays is usually free. But it's very hard to find spaces, especially those that are free, at the weekend. You will waste a lot of valuable time trying to find parking places if you intend to visit lots of places.

I agree with the previous post, you will save a lot of time and hassle by taking public transport to London and for the duration of your time here. Buy a one day travelcard from LGW, use it to travel to central London and then you can travel anywhere you like all day without worrying where to park. Just rent the car when you decide to travel out of London (or have you considered taking the train?).