I've been experimenting with a my cold plate & jockey box rig (I also have a kegerator, and refrigerator to play with).
I agree with @wyzazz the simply placing the cold plate in a mini fridge will not be enough to sufficiently cool your beer for real time service. The first pint or two will be plenty cold, but after that you'll be getting warm beer. Even if the air temp in the fridge is below freezing, air isn't a good conductor of heat, so you'll have a hard time keeping the plate cold when in use. That being said, if you can crank the temp down in the mini fridge low enough to keep a large volume of ice frozen on top of the cold plate, that may work..sort of. Just dont expect a miracle. The cold plate is a heat exchanger. If you run room temp beer through ice in a fridge, the ice will melt and then (as long as the fridge is turned way down) possibly refreeze.
If you already have an old dorm fridge, it's worth a try (with the ice method) if it gets cold enough. It'll be a fun experiment if nothing else. IMHO you'll just be better off investing in the right solution, your original keezer idea, and keep the jockey box for portable fun.
Happy building, and drinking!