A walled secret garden, the Physic Garden is like a time capsule, founded in 1673 to allow apprentices of the Worshipful Society of Apothecaries of London to study the healing properties of plants, it is now a living museum. Tours are offered by volunteers with immense knowledge of botany and the cultural history of the garden or visitors can wander the grounds alone and spot odd and fascinating plants such as the mystical Mandrake.
There are three branches of this South Indian Vegetarian Restaurants across London, two in the West End. They all look pretty unremarkable from the outside, the décor is pretty basic, but don't be fooled - the food is wonderful! It's incredibly cheap and the friendly staff cater for vegans and people with food intolerances really well.
Slightly biased as this is where I live, but Stoke Newington is great place to spend lazy saturday / sunday afternoons getting sloshed with friends. There are 10 + pubs within walking distance who all are great places for one reason or another. I'll pick out the White Hart for its huge beer garden...
It is a music store in Brick Lane. You can spend there the perfect Sunday afternoon, listening to lot of new good music, thumbing through cool books while relaxing in their café. Oh, and the Rough trade photo-booth is a must if you go there!