My hints are as follows:
1)Work out which spurs out of central London will take the longest (and how you can pair them up)
2) This challenge permits the use of Buses and Trains, we know that the current WR holders used buses
3) Check out where the stations are on a street plan, a "Tube map" is no more than a diagram
4) Use all the information you can get at
thetube.com Check especially stations with awkward opening times, stations that have awkward interchanges (I have a huge excel file from somewhere listing the number of steps between platforms)
5) Don't be tempted to take the river boat ;)
6) Note that the person who wrote this is a crazy Northerner, and has only ever been on the tube a dozen or so times
7) Get a working timetable (not sure how, you need a "man" on the inside as far as I know)
(Oh, and Geoff's big tip --> plans (where made) are made to be tweaked (occasionally quite extensively))
Edit: I know that the general protocol is not to give much away, I give this much in order that whoever takes it will have an enjoyable day out. I should like to point out that my personal desire is to attempt and to complete, not to break a WR.