British Museum and Natural History Museum are both amazing, as is the Tate Modern. Covent Garden and surrounds is fun for street performers.
For shopping, the Portobello Road Market in Notting Hill is one of the most famous street markets in the world. The Saturday antiques market is the most popular but there is a street market six days a week. Portobello Road itself is a long, narrow street which stretches over two miles over 6 years ago
Shopping will be high on their list so suggest Top Shop on Oxford Street, Carnaby Street, Camden Market and Portabello Market.
Just zipping around the tube is fun.
Depends on what your 16 year olds like to do. If they like museums, then the V&A usually have some good exhibitions of clothing, I like the South Bank for quirky art displays and off-the-wall stuff. over 6 years ago