You have many opportunities to find cheap accommodation in Hungary, though in some cases you need to know some Hungarian to find the best options.
Tourinform
You can find Tourinform points and offices all over Budapest and Hungary. The signs on the maps and on the streets can guide you to find help.
http://www.hungary.com/
The official website for tourists in Hungary.
Campings
If you have a good tent and a sleeping bag there is nothing else that you need to have a cheap accommodation in a natural environment.
Cheap accomodation in Hungaryhttp://www.belfoldiszallasok.hu/ifjusagi_szallas_hostel?ln=enl
Find cheap accommodation in Hungary for young people here.
http://www.miszsz.hu/msz22.html
This website of Hungarian Youth Hostel Association intends to help you in planning your stay and finding reasonable safe and well managed hostels all over the country.
http://www.hotels.hu/
The biggest accommodation database on the Internet. Search for hotels from budget ones to 5 star luxury accommodation.
Cheap accommodation all around the world
If you travel abroad it is always worth checking these websites which collect the offers of the hostels in any country. In hostels you can usually find cheap shared accommodation in the dormitory rooms.
If you are more adventurous you can try couch surfing. In this system local people or communities offer accomodation. If you choose this option find your hosts. carefully, try to collect information about the people (through social networks for instance) who offer you this option.
http://www.hostelworld.com/
http://www.hostelsclub.com/
http://www.gomio.com/
http://www.couchsurfing.org/
http://www.couchsurfing.net/