you can get a travelodge for around £50/night. a hostel is around £25-£30, but that will be a shared room so isn't recommended. premier inn is likely to be more expensive but is worth checking
if you are going to be in London long-term you should buy an Oyster card and a 16-25 railcard (assuming you are of the appropriate age). You need to ask at an Underground or Elizabeth line station for them to load the discount on to the Oyster card (1/3 off fares, not applicable before 09:30 or 4pm till 7pm going home, so potentially not very useful during your VS but still worth doing for long-term savings).
It's cheaper to use Elizabeth Line than other rail (non-underground services) (assuming in both cases you stay within London).
E.g., Romford is zone 6 and is £5.80 peak, £3.80 off-peak into London including Underground, whereas for East Croydon (Zone 5) it's £8.80 or £6.30 to an Underground station (it's £1.90 cheaper if you don't use the Underground); it's £300 for 6 nights in Romford Travelodge and £373 for 6 nights in East Croydon Travelodge. So e.g., Croydon £16.30/day (cap within London), Romford £11.60/day, something in zone 2 £7/day, all at peak times.
It depends where you are working - if you are in the City (i.e. around Moorgate/Liverpool Street/Bank), then going east is likely to be quicker than other parts of London for the money. However, if you're in the West End then this is not applicable. Also it's worth checking walking distance to train/underground stations. If you're staying in South London then there's often no underground, so you should check train frequencies, as they can be infrequent; however, it's more pleasant to take a train the underground, and if you happen to be working, say, near City Thameslink, then getting one train into there would be preferably to having to use the underground (although as above, it's likely slightly more expensive)
