I havea Canon PowerShot A530. It has a manual mode, is five years old, and has held up pretty well. (Keep it in a case though, I was carrying one around in my purse for awhile and something broke off--thankfully it was new enough that Target exchanged it.)
Amazon has a bunch of used/new ones: http://www.amazon.com/Canon-PowerShot-Digital-Camera-Optical/dp/B000EMU888/ref=sr_1_1?ie=UTF8&qid=1314406697&sr=8-1
STarting at <$90, not sure what the definition of "cheap" is.