i always just try to get lost in a place and find someting local or ask someone where am I going. Since you are in London that should be easy beacuse they speak english.
I would just take the tube to some populated places and just walk around. Leicester square might be cool. I cannot think of anything awesome because i was only in London for a day and a half. Check out some of the Royal buildings they have some great architecture if you are into that oldy stuff. Other than that find a pub get tipsy and start talking with people and you will probably find some really cool places to go or things to do. Or you could just play darts.
g'luck

tate is the usually a good spot... you've probably been there- but theres always new stuff.
i think the hopper exhibition is still going...
and something new for the turbine hall - http://www.tate.org.uk/modern/exhibitions/nauman/ - you might like the sound of it...
also, sattchi has his gallery up and running at the old county hall (by the london eye)...
have a good time.

you should not miss sir john soane's museum:
http://www.soane.org/

I agree with worm....anywhere you are at in this world, especially an urban context such as London, it's much more fun to discover or re-discover spaces and places that are not on a tourist map. Of course the Tate and such places are fantastic. But I found most fun in visiting the places that most people don't.

If you are there on the weekend, go to Camden town....it's a crazy and interesting scene (some ways a bit commercial) yet good fun and cheap food. (+ excellent bargains on things you rarely would look for)

enjoy and get lost!!!!

Avoid camden town at all cost actually. It's a shithole and not worth visit. Except if you want to buy some fresh magic mushrooms. But if you're into bargain hunting, cool clothes etc hang out on portobello market on a saturday or bricklane/ Spitalfields market on a sunday. much nicer.

some suggestions:

take tube to westminster station. arrive at station and admire grimshaw's design. walk out to street level where you have two choices: go west towards parliament and surrounding govnt buildings or walk along thames east which will lead you to millenium wheel and if you've got two working feet, towards tate modern. walk across millenium bridge to admire the city views and end up at tate modern. if you decide to stay in the area take tube to whitechapel gallery where you can visit rotating exhbit and get map for a gallery walk in the area. also recommended northwest is brick lane area where pakistani restaurants galore. the guys who try get you into their restaurants seem sketchy but the food is suprisingly good and cheap. i've been told that jack the ripper once roamed the area in its tiny cobblestone alleys. fun with a group of friends at night!

V&A museum in kensington- lovingly tattered over its life, the musem has great visiting shows as well as excellent industrial design collections. one of the coolest permanents are the rooms with the copies of famous architecture & sculptures. divided into time and place of origin, the pieces are plaster recreations from the gates of hell to michelangelo's david. it sounds lame to see copies of the greats but the scale of the recreations is really impressive and the rooms are packed with them. the room with the copies of famous knights & royalty tombs is super creepy in a la da vinci code mode.

if you've made it to kensington for the V&A, you might as well hit harrods - while people complain of its touristy-ness, i have yet to experience a place like it in the states. the food hall is a great place to pick up small souvenirs for friends and family.

if i you're into interesting clothes/jewelry/accessories especially vintage, i'd rather spend a sunday at portebllo road in notting hill than camden town. not cheap but worth it if you want a special find that you can't get elsewhere.

london's winding roads and expansiveness does not make it easy to find destinations such as above but i do agree that getting lost in its maze makes it all the much more interesting if you have the time to spare.

sorry - i just reread the initial post. you say 'not touristy' so you may consider my suggestions above to be touristy.