Yes for the theatre I think the way to go looking at either the very last minute using the various apps (if you work in the NHS/healthcare both Central Tickets and Tickets for Medics have really cheap special offers albeit often for more esoteric or smaller shows rather than the big west end productions, there probably are similar for other public services) or being very organised and super-early as places like the Royal Opera House have very reasonably priced tickets for seats in the gods but they do tend to sell out quickly. If you don't object to a mild religious bent some churches do free or cheap midweek lunchtime concerts often with a very high standard of performance, and the buildings are often very historic/interesting too.
Thinking a bit more left field, again if you are organised in advance it can be well worth looking outside London at touring or rep shows in the suburbs or towns on train links to London, I've been to shows in Richmond, Croydon, Cambridge, High Wycombe etc at a fraction of the price of similar west end theatres, obviously you have to factor in travel costs and time but if you book advance train tickets it's often still much cheaper (particularly if you want to eat out before/after the show or go for a few drinks somewhere local). I really love the annual Shakespeare festival in Cambridge, tickets cost buttons and the train only takes 45 mins or so and costs about £10 for a super-off peak return...
I've always enjoyed free (well, you are asked for a donation/voluntary contribution) comedy at places like the Angel Comedy club, obviously it's not big names and it's small rooms usually above pubs rather than proper theatres but you can find hidden gems. Also if you know where to look comedians will sometimes run trial gigs of their work in progress at a much reduced price - the Arts Depot in North Finchley is an example of a theatre used for this purpose.
Obviously there are all the galleries and museums that are totally free or very low cost entry but it bears repeating because there are so many of them in London, I could spend weeks going over the entire collection of the British Museum or the National Gallery or the Tate Modern for instance. Plus there are the smaller, quirkier ones - search old threads for ideas but my personal favourites are the Sir John Soames and the Wellcome Collection. Places like Somerset House also often have small temporary exhibitions spaces with interesting exhibits, perhaps not enough to base a full day out around but combined with a nice walk along the river or similar it's great.
For visitors to London riding around central London on double-deckers but using regular TFL services rather than the expensive tourist buses is a great way to see the city, I think it's the C2 route that's often recommended that takes in loads of london landmarks.
A good experience is to go and see a TV or radio show being recorded at ITV studios or radio house, you usually have to enter a bit ballot so it's a bit of a lottery, plus lots and lots of queuing and waiting around but I've been to see Last Leg, Have I Got News for You and Graham Norton previously, all a good laugh and I don't think we had to pay anything for the tickets ...
When the nicer weather rolls around it always surprises me how many nice open public spaces there are in and around London and a great low-cost day out is a long walk around a park with a picnic - the obvious ones are the central london parks and places like Hampstead Heath, Greenwich etc. but if you use the Komoot app it's great for suggesting much more off the beaten track walks within or around the M25 belt.
Food I'd struggle with a bit more as I don't know many places you'd truly class as a cheap but excellent meal in London any more, prices are high across the board, even 'street food' at the likes of Borough Market or Covent Garden ends up being quite fancy and costing a bomb often but I do quite like going on a adventure to find London's less well-frequented cuisines, e.g. a trek down to New Malden to Korea-town or Wembly for Sri Lankan and South Indian food (I work in an extremely multi-cultural team so always asking for recommendations for where people go for 'home-style' food). Or for something fancier, many of London's high end restaurants do a reasonably priced set lunch menu on weekdays, obviously paying £75+pp is not most people's definition of cheap but given you'd often pay twice or more to eat a la carte from the evening menu at the same restaurant I think it's not bad value for a special occasion!