Stroll through the Embankment Gardens.
Spend some time in the little garden at St. James Picadilly and be delighted by the memorials.
Visit St. Paul Covent Garden to see the memorials to great British theatrical personalities.
Tour the Globe Theater and see a play there.
Book a performance at Royal Albert Hall and eat a picnic supper in Hyde Park beforehand.
Have a wonderful time.

See whatever special exhibition is on at the Tate. We saw Olafur Eliasson's The Weather Project, which was incredible, but I bet they have other great projects there as well. Enjoy the rest of the Tate while you're there.

(Can't believe museums are free in London--wish they were everywhere!)

The British Library near St. Pancras, including a guided tour; and I second The Wallace Collection at Manchester Square - a great little museum!

Go and see WEST HAM UNITED play Football (soccer to the Americans)

I'm with Tom v here: Go to a football match (but not West Ham for God's sake, get yourself up to White Hart Lane for the real thing)

In the summer go to a Cricket match at Lord's (no booking necessary apart from the internationals), go and look in the cricket museum there.

Some more off-beat ideas from a Londoner:

A brewery tour (Youngs in wandsworth is good, and still has horse drawn drays)

Go to the National Gallery and buy a copy of Constable's view from Hampstead heath, then go to Hampstead Heath and compare the two - it's instructive in many ways.

While on the Heath go to Kenwood House to see the Iveagh Art bequest, and if you're lucky there nmay well be an open air concert in the evening.

In winter go ice skating at either Broadgate or Somerset House.

Visit Apsley House (wellington's home.)

Go to Bermondsey Market (opens at dawn) for antiques and art.

Go to a greyhound track (Wimbledon or Walthamstow)

Check out a traditional pie and mash cafe. Try the jellied eels, if only just to say you have.

Get a "ruby" (ruby murray = curry) in brick lane. This is where the "ripper" walks end so you could combine the two.

Go to the RAF museum in Colindale - great for little boys of all ages.

Go to the Candlelight concerts at St Martin's-in-the-fields.

I'll come back when I can think of some more.

After declining invitations from Tony Blair and the Queen have lunch with Ben Haines.

I'll add:
Geffrye Museum for its furniture and textile collections
St. Bartholomew the Great for its architecture ( and church services)
a walk by of St. Brides to see its wedding cake steeple

Marvel at the reading room in the British Museum

Cross the Thames on Westminster Bridge and visit the Florence Nightingale Museum, the Museum of Garden History, return by the Lambeth Bridge and visit the Tate Britain.

Travel to the Dulwich Picture Gallery

Try to find nightingales in Berkeley Square

Walk the Island Gardens/Greenwich tunnel

Admire the artwork on Bayswater Road on Sunday

-Browsing at Harrod's
-Fish & chips

ttt

Whitehall Palace Banquet House
Cabinet War Rooms
Ceremony of the Keys
Highgate Cemetery
Portobello Market
Walk around looking for Blue Plaques

purchasing baguette sandwiches from a local baker and having a picnic in whichever park is closest
try to visit all the locations on the London Monopoly game board
take a boat trip from Hampton Court Palace through the Teddington Lock to Richmond
take your picture straddling the Prime Meridian in Greenwich
National Gallery and National Portrait Gallery
making brass rubbings in the crypt at St. Martin in the Fields
cathing a train at Waterloo Station and watching all the arrival and departure signs "flip" to the latest schedule then racing to your platform (although I think all the arrival/departure signs are now computer screens)
browsing in the Accessorize stores (great, fun jewelry, etc) -- nothing that I know of like it here in the States
and on and on and on!
Annette

Go to the Royal Mews to see the Royal Carriages and the horses

Theresa

Here's the one that I always add - bring nuts to feed the squirrels in St. James's Park. This is best done before 10 a.m. or in late afternoon. They are so tame that they eat right out of your hand. We were near exiting the park last year and I had given all the nuts away [sometimes to children who had nothing to offer]. One little squirrel looked deeply into my face as though he couldn't believe I hadn't saved one for him. Then he practically climbed up my leg - a bit scary as I was wearing a skirt. We are the only people I know who fly nearly 7,000 miles each way just to see squirrels. The ducks and other birds are also very entertaining as are 100's of other things in my favorite city.

Some more that I have thought of:

Get one of Andrew Duncan's walking books and follow a trail.

Go to the Inns Of Court and get them to let you in (they will if you ask nicely and they're well worth seeing.

The Masonic Temple in Gt Queen St is also open to (polite) public and is well worth a quick look (there's a masonic museum there too - but that's for those who are interested in apron related activities).

Dulwich picture gallery repays the effort.

Try "Village london" eg Hampstead, little venice, Kew and thereabouts etc.

Go to the sing-a-long sound of music.

Try out an off-westend play in a fringe venue.

Go to a "school dinners" disco. This is best done roaring drunk.

Go to the bethnal green museum of childhood.

Get a real breakfast, and a pint, at Smithfields meat market.

Buy flowers at columbia Rd market.

Go evening horse racing at Kempton Park. (summer only)

Get the Tate-to-Tate boat along the river.

The Thames Barrier is much more interesting than it sounds.

Watch a trial at the Old Bailey

Go into a neighbourhood pub and talk to people.

Get an englishman to explain how to make tea properly. You'll never touch coffee again.

Go and look at the Pelicans in St James' Park.

More when I think of them....

David West: What is a "school dinners disco???"

What is a "school dinners disco"?

In short; carnage!

It's a disco in which every one puts on their old school uniform (or very approximate versions thereof) and has a raucous evening dancing to the hits of their youth, whilst being very very drunk. THis will give you an idea....

http://www.schooldisco.com/SchoolPhotos/default.aspx

Lunchtime concerts at the City churches.

Attend free lectures.

Walk aimlessly in the Square Mile,check out new buildings.

Go to Borough Market on Saturday morning.

A really good curry on Brick Lane... YUM!

Go to Waterloo and get the train to brighton.