I am in no way suggesting that 3 days is sufficient time to spend in London. I would suggest you spend as much time there as possible. In three weeks, I have commuted back and forth from Chesterfield to London for no longer than two days however that has been purely work related. My boyfriend and I did manage to do a weekend getaway to London during a European trip two years ago.
Arrive in London at a reasonably early time. If you have quite a bit of luggage with you, store it at reception of your hotel so you can be free to do a lot of walking around and exploring. Now the first thing I did was go to Harry Potter World because I am such a huge fan and it was my dream come true! If you're not though and it is your first time in London I would suggest doing some sightseeing. Go snap a photo of Big Ben, take a bus tour (you see heaps and it is a good way to get your bearings) and maybe even do the London Eye.
If you love a good show, you have to see a musical on the West End. I have seen three musicals in London; The Phantom of the Opera, The Book of Mormon and Matilda. I would recommend all of them but Matilda was definitely the most outstanding. For shows that have been out a while, you don't have to pre book them in advance, there are heaps of small stalls to buy cheaper tickets the day of. If you want to see a more popular show you probably won't be so lucky. Dinner and a musical is my ideal night.
The next morning we went to the Sky Garden for breakfast. It is such a beautiful spot overlooking the city and so well priced considering the venue. The menu is small but delicious and walking around the gardens is a lovely way to start the morning. Hopefully you go on a morning that isn't too foggy as that will completely cover the whole city up. After that head over to another activity. We went to Madame Tussauds because I had never been and wanted to see what the fuss was about. You can also visit the London Zoo, the aquarium or I would highly recommend going to the markets. I have only been to Camden markets but it was such a good atmosphere although it is extremely busy so if you don't like crowds it won't be for you. Actually London as a whole is probably not for you if you want peace and quiet.
That night we saw a movie in Leicester square. If you are a movie buff, London has such a great movie culture with a massive IMAX cinema and some really cool indie cinemas too. If that's not your thing, go and experience the nightlife. It's the UK so there's a 10 pubs per street and plenty of other bars and clubs.
The next morning we took a tour of Buckingham Palace which was again a dream for me. Even if you don't do a tour you should go and see the Palace from the outside or watch the changing of the guard. The gardens are nice to walk through as well. After that we headed back to the airport and the weekend was over!
It can be really hard to find accommodation in London on a budget. I usually use Booking.com to find cheap deals often at the last minute. If you are on a tight budget, consider hostels as well where you can always get a private room if you aren't comfortably staying with strangers. If you are thinking of signing up to Booking.com, use my referral link to get 5 pounds cash back after your first stay https://www.booking.com/s/34_6/0642415d.
(Hint you can sign up using different email
addresses to get money back each time!).