First of all, to find all the things you can do totally for free, check out my designated post for those. It was simply too much to pack all of it into one text, so sorry for the inconveniences.
Here we go now with some spots you can visit and ideas you can do, that are not completely free, but will help you save some money as they are aboslutely the cheapest options.
In case you are planning on visiting some of the musicals while you are in town, it is worth downloading the app Today Tix. It offers last minute, special offer and simply low cost tickets for the most famous shows. Tickets start from 15 pounds for regular musicals, while pieces at The Globe Theatre are even cheaper: you can get standing spaces in Shakespeare's masterpieces for as little as 7 pounds per ticket.
Some musicals also offer on-the-day tickets when you queue up in the morning. These are often fantastic seats for 20 pounds, or in the case of Mathilda (15-25 years only) even only 5 pounds! Check the websites for additional info on age restictions and opening times of the box offices.
Once you are making our way around London and are searching for a cheap place to eat, consider going into some supermarket. The huge chains like Sainsbury's, Tesco or M&S offer Meal Deals. Here you can get a sndwich, some snack and a drink for 3-4 pounds per menu. This is a fantastic option to save some money during lunch and rather eat somewhere really nice in the evening then.
When travelling around London it is highly recommended to use an Oyster card or your own contactless to pay for all your rides on public transport. These have, for example, a daily maximum and weekly maximum, so that you automatically are no longer charged for your rides as soon as you reached the respective price. Still, try to take busses rather than the tube, especially in the city centre. Yes, I know, they take much longer, but you can also see plenty more and it is less than half the price (1,50 pounds/hour).
A pretty cheap and fun option to go around the city are the Santander bikes. You can get one at the many, many stations spread over the city and leave it at any other one you like. The first 30 minutes are free, and you only pay 2 pounds after that
The tickets to most of the top attractions are relatively expensive. Things like the London Eye, going up the Shard or entering the Tower of London are not very cheap. One thing, howwever, is: Going into the Tower Bridge and crossing the glass walkways with the cars and the river Thames underneath your feet. My advice: Book your ticket online already. It is much less queueing then!