When I was a student I used to stay at youth hostels - and there are plenty of those in London, see www.yha.org.uk for further details. You need to join the Youth Hostels Association in your country in order to stay at YHA hostels in other countries.

I appreciate that none of these hostels are close to LHR, but if you wanted to check in in the morning there is always the option of getting a Travel Card, taking the tube to LHR and then going back and spending the rest of the day sightseeing. (BTW, are you sure you CAN check in in the morning? I've been in many places where check-in desks were not open until a few hours before departure - typically four hours for long-haul, two hours for short-haul)