I know there's professional grade noise damping material but if I'm on a budget, what other options are there? I've heard of denim, roofing shingles and just cloth in general. But what other options are there for the budget foley room?
If you have 70 dollars to spare, you can get 6 2' x 4' uncovered Rockwool panels. These are huge, and it's a lot of them. All that's left for you to do is wrap them in cloth and frame them. The difference between this option and a 20-30 dollar solution is far more than the 50 dollars you might save.
The following is what I purchased for my studio: http://www.atsacoustics.com/item--Roxul-Rockboard-60-Case-of-6--RB60.html
Make sure you read up on what the needs will be for your own room. You may need a less or more dense fiber.
I'll second Iain's comment. I wanted to save a few bob by making my own panels for my studio. They're not tuned to any frequencies as I simply wanted to deaden the reverb in my 4m2 room. Happily turned out really well and my work translates rather splendidly to larger "proper" mix studios.