The Soane Museum has to be seen to be believed. In fact it is really tricky to describe the place in a way that sounds interesting, but believe me, this is a fascinating museum and one of the great free things to do that London has to offer.
Soane was a prominent architect who designed the original Bank of England building. During his lifetime, he consciously made his house into a museum of architecture, art, and sculpture, obsessively collecting objects from around the world and contriving ways to display them in his house. Soane was a master of light and illusion, and though the house is of a fairly modest size, even the smallest rooms seem huge, and every nook is filled with sculpture or architectural elements brought to him from around the world. His collection is truly stunning and even includes the sarcophagus of Pharaoh Seti I (which he bought for £2000 after the British Museum refused to pay for it).
The best part of the museum is the building itself: this is an undeniably eccentric and atmospheric place, including a crypt for his wife's lap dog that simply reads "Alas, poor Fanny!", a basement room for an imaginary monk named Padre Giovanni, walls that fold out to reveal further paintings behind or to let light into floors below, and intricate glass domes for even the smallest cupboard of a room. Once a month they open the museum at night for candlelight tours, and I would love to go back for one of these evenings.