Depends what you like doing, but you could cross Westminster Bridge (next to the Houses of Parliament+Big Ben), walk along the South bank of the Thames after going on
the London Eye,(checking out the Dali, and Saatchi Galleries too), via the Festival Hall ( excellent 1950’s Festival of Britain Architecture), the Hayward Gallery and the National Theatre (60’s/70’s Brutalist Concrete).
Get the best (ground level!)view of London from Waterloo Bridge, where, looking North East to the City of London, you will see 350 years of Architecture from St Pauls Cathedral to the new ‘Gherkin’ (Swiss Re Building).
Carry on along the south bank to the Tate Modern art gallery (essential), and then past the new pod-like London Assembly to Tower Bridge and the Tower of London. Take a southern loop along Butlers Wharf to the Design Museum, then back to Tower Pier to get the Boat down the Thames to Greenwich, where you can see the Observatory+ Meridian Line, and get some excellent and cheap noodles at the Chinese Noodle bar on the corner by the Cutty Sark ( an old Tea Clipper in dry dock). Cross the Thames from Greenwich in Brunel’s ancient underground foot tunnel, and catch the Docklands Light Railwayback to Tower Gateway. Go to St. Katherine’s Dock nearby and look at the yachts.
If you’re young enough, and have serious amounts of metal in your face, you’ll also enjoy Camden Lock Market on a Sunday morning.
You also can do weird things like discover the old surgical museum, with it’s public viewing gallery, near Guy’s hospital at London Bridge, or try to get into the Old Bailey to see a criminal trial, or visit the underground London Silver Vaults, but that might be a bit specialised!
There are also organised Walks with various themes that any Tourist Office should know about. Walking round the old parts of the City of London at weekends when it’s almost deserted is fascinating - I also love the Barbican, a kind 60’s ideal development of offices, apartment towers and Cultural centre with Theatre,Cinema, and Galleries in a concrete retro style built around gardens and pools. Also the Museum of London nearby is worth a visit.
Depending on where you’re staying, Londoners generally go out more locally, leaving the Centre for tourists especially at weekends - unless you’re seeing a new movie, or need to go to a famous restaurant, or something. If you’re North of the river, Islington or Hampstead are busy places at night with a lot of places to eat and arthouse cinemas etc, Richmond in the South West is also very pleasant.
Of course if you want some Heavy clubbing, or have to be seen in ultra-hip Bars rubbing shoulders with celebrities - I’ll leave that to others to fill you in !
Looks like the summer’s finally arrived too.