Greenwich Marine Time Meuseum & Observatory,in Greenwhich,London. Its the place where u can see the 0 degree londitude which is used for world time calculations, you can stand on either sider of the globe, the place has historic and scientific importance and the best part is it is that the entry is free of cost. The place also hosts a fair/flea market from where you can buy a lot of traditional stuff. Lovely small place to be in.
Other places like Westminster Abbey,Towers of London, A cruise on Thames, A musical can all be enjoyed. If you to westminster abbey - which is a church - try to attend to one of the prayer services, you could be a part of the royal choir that sings the hyms there.
I can write a lot about the places to see there, but the above two were my favourite.
You can also check www.londontown.com for more information.
Happy London visiting! A great place to be in.