My kids loved the British museum, especially the egypt stuff. They hate modern art so for us, the Tate is not great but it is a wonderful gallery. And walking across the millenium bridge is cool. I love the National Gallery, my faves being the impressionists gallery and after visiting a few times, the kids are starting to recognise styles etc.. The map they give you, highlights the "big" paintings so you don't have to wade through a lot of boring art (sorry art lovers..)
The Natural History Museum is ab fab, but be warned the human body exhibit although very extensive and quite amazing, begins with conception and is a wee bit graphic. My kids (11 & 14) loved that museum.
We also did a secret walk last time we were there, visiting neat little churches, which was fun. We had a written guide, which we followed.
http://www.london-walks.co.uk/29/index.shtml Walking, walking, walking. That is our favourite thing to do as a family in London. Come up at picadilly circus, it will make an impression!!!
If you really want to wow them, if they're into fashion, walk up and down new bond street and look in the windows of the designers' shops.
We are specialists in travelling on the cheap. even in London!