Not a local anymore, but here are a few suggestions of somewhat less touristed places that you might not have visited yet.
1) Have you been to Spencer House? It is only open on Sunday. They do an excellent tour
2) Are you a foodie? Try the Maltby Street Market on Saturday morning
3) A couple lesser known sights that are interesting are Dennis Severs House and the Geffrye Museum. See if either appeals to you.
4) See an interesting film at the BFI Southbank or at the IMAX theatre in the middle of the traffic circle by Waterloo Station
5) Visit the Kensington Roof Gardens http://en.m.wikipedia.org/wiki/Kensington_Roof_Gardens
6) Finally, don't overlook all the various walks offered by London Walks. Some of my favorites have been walks through lesser known neighborhoods. I always learn something new from the guides (like how the most important dates in English history all end in "66" ... 1066 - Norman Conquest, 1666 - Great Fire of London, and 1966 - England wins the World Cup). :)