Your question is a loaded one and cannot be simply answered with a specific camera and housing. Some digital still cameras take video clips which are not the same as a video camera. Some video cameras take still photos. "Cheapest" can mean crappy pictures which is money wasted. You will have to first decide on the type camera you want to buy, still or video, then go from there. Housings often cost as much, or more, as the camera. If you buy an expensive housing that is camera model specific then the housing will likely be around much longer than the camera, especially if it floods. You can focus on the latest model which is only the latest for a few months, or you can stay behind the curve with an older model. You will have to do your own research and there is plenty of answers on this forum.
It sounds like you don't want to spend a lot of money. You don't have to and can still get a good camera.
Here is just one example if you decide on still digital:
You can get a refurbished Olympus C4000 (4 megapixel camera) online for around $320. and a new Olympus PT-010 housing (no longer mfg.) on eBay direct from a seller in Japan for $179. Add shipping to both. Refurbished does not necessarily mean overhauled. It can simply mean re-packaged after a customer brought it back due to buyers remorse. They have a 90 day warranty. You can spend a little more and find a refurbished Olympus C5050 and get the PT-015 housing. I'm sure there are similar deals with Canon cameras.