Random Tube-Travelling Day is something I've been meaning to do for years now. Assign every tube station on the system a number. Find a way to roll that range of numbers with dice. Start at one of the central stations, roll for a station and go to it. At each one, get out and take pictures (and maybe post where you are with sms-to-lj so people can track your progress). Possibly do other things, like 'buy something' and 'talk to someone'. Roll another station. See how many you can visit in something like 12 hours. Finish with drinks at the nearest pub to the London Stone.

I like your suggestions, and actually it'd be more fun if lots of people were doing it at once (if it were possible to persuade people to spend a day on the tube, which I doubt). I don't think you could get away with leaving a trail, though.