So, I have had a lot of fun with cheesy little chess computers. I had one when I was very young. It talked and had wooden pieces with magnets. I played it on road trips for hours.
A few years ago I discovered how to beat all of these little chess computers with the horizon effect. I'm not talking heavy database run engines, just most of the cell phone games, handhelds, and laptops that have sprung into existence in the last 20 years.
I've found that I have to look for moves that are bad or weak moves but allow for a stronger continuation. Allow pawns to be doubled, open files at great cost to king safety, or allow an annoying pin. The computer will have not considered such terrible moves, and the lines that develop are unexplored by the computer. In those lines you can frequently find mating attacks and material gain overlooked by the computer. This is your compensation! It becomes amusing as you look at a position and realize you get a free material or a discovered check that completely demolishes the AI's game.
This is just something I've observed with weak computer chess games, handhelds and the like. I hope if anyone has become frustrated with one of these that this type of thinking will produce better results.
It'd be cool to hear of any other peculiar tricks or processes that produce good results against some of the chess AI's.