I'd recommend the North Eastern, Isaan region of Thailand. Here everything will be as cheap as you can imagine. Plus you'll get a real cultural feel of the place where not so many tourists venture when visiting Thailand. Bus travel is cheap and a 10 hour journey will cost you about 400baht. Food is delicious and cheap, and budget accommodation won't cost the world, however it varies in standards.
If your looking for the beach, Koh Samet is probably one of the cheaper options and this is where many local Thais take their weekends and vacations. It's also close to Bangkok so you won't have to spend much travelling there!
However the typical trodden path to the south can also be very cheap if you spend a little time looking for budget accommodation and are willing to eat street food often. On Koh Phangan and Koh Tao for example you should get rooms for about 250baht a night. It'll be basic but you can spend your money on better things, like scuba diving! In these places however there are many tempting, potentially expensive ventures to spend your money on - be warned!
I hope this helps. If you need any more ideas just ask.