All these are mostly touristy but still worth it and fun:
A stroll through Hyde Park to the Serpentine waters, and, if they still have it at this time of the season, hiring a rowboat for a peaceful late summer bit of boating!
The London Eye, apparently on a clear day you can see all of London and beyond to the surrounding counties.
I second Camden Market -- fun, colorful, interesting things to browse and buy, interesting people-watching.
If you do Hampstead Heath, walk up Highgate Hill to The Flask pub -- hundreds of years old, great atmosphere, famous clientele (Michael Palin, Terry Jones; additionally Sting lives across the street).
Tower of London if you and your visitor are into history.
A drive out through Epping Forest, on the eastern border of Essex, very accessible from east London. At this time you may get the start of the leaves turning color. Dick Turpin used to hide out here, and Queen Boudica was said to have had either one of her last stands against the Romans in the forest, or one of her final encampments.
Again for history fans, at the Chingford edge of the forest is the 16th century Tudor hunting lodge built by Henry VIII and also used in turn by Elizabeth I.
Drive north from there on the Sewardstone Road and visit Waltham Abbey, very long history -- there has been a church there since Saxon times, then the original Abbey was built by King Harold II, the king killed by the Norman invasion at the Battle of Hastings. His body was said to have been brought back to Waltham Abbey and buried here. King Harold Day festivities are held October 17; the town has fun with lots of activities.
There are also some nice pubs in the forest to stop at -- the Robin Hood, serving Thai food!, The Owl at High Beach (or Beech) nice place.
