Although London has some very pricey attractions, it also has plenty of free ones. These include:
Science Museum (great for 'big kids')
Natural History Museum (moving dinosaurs!)
Victoria & Albert Museum (superb fashion collections, and much more)
Imperial War Museum (not just military exhibits. Lots of stuff about the lives of civilians during the Blitz as well)
National Gallery (Stunning masterpieces)
National Portrait Gallery (Check out the great photography section)
Tate Modern (Modern art in a stunning setting)
British Museum (Mummies!)
All of those galleries and museums are massive and take ages to go round. (I've dragged unwilling kids into the Science Museum. Five hours later I've had far more difficulty in getting them out of the place!)
There are also other free attractions, like Covent Garden Piazza. This area is packed with bars and restaurants (across all price ranges), alongside some great street entertainers. (Jugglers, acrobats, folk singers, opera singers . . . They're all there).
Around the fringes of Covent Garden you'll find some good quality, fairly cheap bistro-style restaurants. This is my favourite:
http://www.bistro1.co.uk/
If you want to see a show or play, and don't mind only having a limited choice, head to the 'Tkts' booth (in Leicester Square) which sells cheap tickets on the day of the performance:
http://www.officiallondontheatre.co.uk/tkts/le icestersquare/
With a bit of forward planning, you can have a cheap weekend in London, packed with interest.
Chris