What to do in London with no money?

Tate Britain, Tate Modern, National Gallery, British Museum, London Museum, Natural History Museum, V&A, Science Museum, Serpentine Gallery, White Cube, Royal Parks, Southbank.
And on and on, all free.

Comedown to the Trackday Sunday and meet some people.

+1
art galleries city-wide, almost all free. cork st, viner st, hoxton, plus all the big ones.
borough market... mmm
camden market (if that's your thing)
there are usually plenty of free gigs, you just have to do a bit of (internet) leg work and risk an unknown being shit.
parks.