Getting lost is an age old suggestion but a good one. Put away your cell phone or your A-Z and just wander around. I still do this regularly even though Ive lived here for nearly 10 years. London is a big city and it drip feeds you its secrets sometimes. You will come across things you wouldn't usually, in amongst the Starbucks and pret a mange there are some amazing sights. Just 'being' somewhere is a great experience and one no guide book can help you find.
Also the Queens back garden, these beautiful grounds are great for a Sunday stroll to see the pelicans or to feed the squirrels. Duck Island Cottage is worth a look too, a idyllic little cottage situated in the grounds which was specially built as the bird keepers home.
The tunnels are a Crane.tv favourite. We have used the space as a backdrop for several films including a music session with The Civil Wars. The tunnels have a great story behind them and feature one of the only legal graffiti walls in London on their Leake Street side which was championed by Banksy back in 2008.
There are many quirky book stores in London, but I really love the selection that Daunt Books have, they often have bespoke editions which you won't find in other book stores. Also, the people that work there are so helpful and knowledgable. I can never stop myself from walking in!