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.
Just down the road from where I live is this Art Deco cinema where mainly locals go. Watching films and sitting in this old theatre really adds another level to the experience.
Signature cocktails like "concrete", with actual chalk on it, food is good and the places has a small backyard garden. Recommend to try the food, as they are small tasty plates.