Speaking to people who have lived in London a while it's easy to take the city for grated after a while. I'm determined to remain a tourist as much as possible especially when so much of it is free!
Most of London's museums and galleries are free (at least for standard collections - you can expect to pay for exhibitions and it's always good to give a donation), that said living in London means you don't often have time to go and it the weekend you want to keep out of central to avoid the tourists! Enter Lates! Nearly all the biggies are open late a night or two a week.
Mnay hold special events like sleepovers, silent discos etc (look out for them I've not been but they look seriously cool). So far I've just been to the National Portait Gallery Late Shift. They have a bar and a dj (who was playing Alt-J so I was more than happy), it gets pretty busy. You also get the booklet above by Marion Deuchars. Much like her activity books it's like a school exersize sheet encouraging you to draw things that inspire you as you walk around.
I saw the Road to 2012 exhibition - photographic portraits of contributors to the olympic games - and I highly recommend it. I especially liked the Local Heros section with portraits of East Londoners.