Hope you'll forgive me for starting two London threads in quick succession, but I'm spending a fair bit of time there over the summer.
Things I have done, like and would happily do again:
Wellcome Collection
Geffyre (I think this has just re-opened?)
Hunterian (still closed, I think)
John Soanes
Walking from St Katherine's Docks, through Wapping and Limehouse and checking out the old pubs
Highgate Cemetery.
Kings Cross to Camden or Angel along the canal
Primrose Hill (not really off the beaten path but very lovely)
William Morris house
Greenwich stuff, and walking from Greenwich to Deptford
Anything either pretty, or with good historical interest would particularly appeal!