For a budget hotel in a central location my favourite area is Bloomsbury, easy to get to from Kings Cross or Euston if arriving/ departing by train and walking distance to Covent Garden, Soho etc but quiet at night if you are off the main roads.
Try the hotels on Cartwright Gardens - (George Hotel, Crescent hotel ,etc) or Gower Street (Ridgemount, Arosfa etc) or at somewhere like the Bedford Hotel Russell Square (bigger hotel so usually have rooms if the others are full).
Bloomsbury has easy access to tube, buses and the West End is within approx 20 minutes walking distance.
www.georgehotels.co.uk/Rooms
www.crescenthoteloflondon.com/Rooms.asp
www.harlingfordhotel.com/booking.html
(there are another 3 or 4 on that street)
Gower Street (get a room at the back as it's a busy road:
www.ridgemounthotel.co.uk/room-information
www.arosfalondon.com/
Bedford
www.imperialhotels.co.uk/en/bedford
Almost any of the small hotels on Cartwright Gardens, Bloomsbury would be my first choice though, (George hotel, Harlingford hotel, Crescent hotel etc - all small, family run hotels with full English breakfast included); Also Celtic Hotel behind Russell Square tube, some of the Grange Hotels (small chain) in South Kensington, Bloomsbury (Bedford Place) and Westminster - particularly the Grange Lancaster "semi en-suite" rooms (shower in room, shared toilets in corridor, from £46 (Sunday) to around £60/£70 (weekdays) incl breakfast), most of the budget hotels on Gower Street (Fitzrovia/Bloomsbury) eg Ridgemount hotel, Arosfa hotel, Jesmond hotel, Both 72QT and Number 63 just north of Hyde Park, Luna & Simone or Lynton in Victoria/ Belgravia, Kip Hotel Hackney (20 mins on train to Liverpool Street or train to Highbury & Islington then change to tube, station is literally next door, also a nice bar and live music next door (Oslo, Hackney), some of the Premier Inn Hub hotels if you are booking far in advance, but even then probably not on a Friday or Saturday night (better value elsewhere then).
On a really low budget - some of the small hotels near Kings Cross are fine, but basic - eg: Meridiana, Crestfield, Angus. Kandara Guest House (Highbury & Islington).
Always book direct for best rates (booking sites charge the hotels up to 20% commission) and often sometimes booking direct gets you extras (like breakfast) included for free which you don't get by booking via third party sites. Booking direct with the hotel also means all the money goes to the hotel rather than 20% of it going abroad to an American corporation, so better for the local and UK economy.
Friday and Saturday nights are the most expensive, Sunday nights are the cheapest, followed by Mondays then midweek.
Of the budget chains easyHotel is probably the worst, Tune who are similar are slightly better (but charge extra for towels and wifi etc), Premier Inn is probably the most consistent/reliable 'budget' chain, if rather corporate/ bland but their dynamic (demand based) pricing model means rates are often not really that "budget". Same applies to their Hub hotels. The Hubs near Covent Garden, Westminster and on Brick Lane are all good. Agree that the Blackfriars Premier Inn is also a great location, near Borough Market, but quiet.
Best of the budget chains is probably Motel One - although again their rates are not always that cheap. Of the hip/trendy/boutique budget chains Qbic, Citizen M and Z Hotels are all decent, although very small rooms at the latter. Sign up for their mailing lists to get regular offers (permanent 10% of at Citizen M for 'citizens' (cheesy term for people who sign up for their mailing list), frequent 20% off offers at others. Hoxton hotels also do offers from time to time, although seldom at the bargain rates they used to do....
In summer many of the student residences do decent priced but fairly basic accommodation, usually single rooms. YHA private rooms also an option.