Corsair, NZXT, Fractal Design and Nanoxia all have good, cheap cases with cable managment, the Antecs over about $60 are good too. While case design helps, I think cable management comes down to your skill and creativity with a zip-tie. Also a semi- or full modular PSU goes a long way.
I'd got for a Nanoxia Deep Silence. No OC so you don't have to worry about heat and that case should be whisper quiet, but it is boxy and plain.
It's a bit hard to pick just one case for under $100 because each have pro's and cons. Pick a few you like then ask for recommendations from that selection.