There is sooooo much to do and cheap cheap!!!!! Start your morning on a cruise on the thames up to greenwich (take a picnic basket with you a small one) (prefferable the tour ones as they give you great history info ) Stop of at greenwich they have a big old looking pirate type shop/boat there and if you walk up to the top of the hill it its where 0 lonitude meets 0 altitude(only place in the world) you can take photos where you have one foot on each side of the line. Great views from the top of the hill. You can have picnic there in the park.
In london on the way back , stop in at covent gardens, great preety markets there and a yummy pasty shop near by. They normal have people doing shows like clowns stuff like street theatre. Near there they have a b @ 1's get in for happy hour 2 cocktails for the price of one! They have a booklet of cocktails.
Other thing to do london dungeon ( i like it , but im a big kid) about jack the ripper, the plague people jumping out at you , like a horror house.
In covent gardens need to go and find "neil's court yard" hippy closed in tea place with different colour walls and bead shops. When its winter i used to go there all the time as it was bright.
For dinner try some morracan at covent garden place called souk i think, has belly dancing.
Go and see a show they have so many!!! You might be lucky and get cheap tickets if you want to take your chance you just go up to the theatre and see if they have cheap tickets, because the day ones they sometimes sell tickets cheap 5 mins b4 the show jsut to sell it. ( i watched guys a dolls with ewen mcgregor for 20 pounds!)
Harrods it always good to go see if you havnt been there, traflgar square climbe on the lions! and the theatres are near there some on the strand and others near shaftsbury avenue.
Free things - hyde park, st james park in front of buckingham palace, i see people who do horse riding around there not sure how much that is though.
there is a great magazine thats free at the main tube stations comes out on mondays forgot what its called, but has everything in it. Has a guide for mon to friday for clubs , diff types of music genres and what gigs are on, fun things to do and special offers. its great get it!
Also a must go to the pub "ben crouch" awesome scary pub in the city on wells st i think, try and find the bathrooms its funny!