I really don't think there's any way you could feasibly make an embedded system that could play flash games, at least not for a relatively simple project. Flash runs under windows, so you would basically have to create a windows environment in your device.
You COULD, however, do it if you used an actual computer as the basis for your project. not sure if that was your original plan.
but then it should be somewhat easy, you would just want to build a small computer and the main challenge would be interfacing controllers with the flash game. if you made your own flash games, perhaps you'd have be able to do that. Or perhaps you might want to use some other game format besides flash.
building a small computer these days isn't that hard. if you look up the line of EPIA mainboards made by VIA, they are about 7 inches square with built-in processor, video, sound, etc. and accept standard RAM, hard drive, etc. and you can use a very small power supply with them.
I am currently building a small system like this to go in my car, and the computer case is smaller than a sheet of paper, and about 2" thick (drives are mounted externally)
so that would make a reasonable game console... however it would cost around $500-600 for all the parts you would need I think (motherboard, RAM, controllers, small LCD display, drives, etc)
anyway, just some food for thought.