You probably could. Is this the arduino you're talking about?
I'm foggy on the details, but that is not straight up compatible with the current Arduinoboy code. You should use a pro mini instead, which doesn't have the USB port but is cheaper and compatible.
There's kits on ebay intended to use for an internal arduinoboy. It has all the circuitry you need for an arduinoboy (besides jacks and stuff). I'd probably do that as it will be the smallest choice.
What are you going to do with the gameboy cable? Cut off the USB cable?
Do you currently have one of the midi cables you talk about on hand? I have one and it's a little unwieldy. Just super long, not that flexible wiring. Easily tangles.
Oh and you'd have to make sure you use a link cable harvested from a 4 player adapter, or modify an aftermarket one to supply 5v from the gameboy.
What about the LED's? How will you mount them in the plastic? Those cables have 3 LED's already, but you need 3 more. Same goes for the button. Wiring will quickly eat up any free space.
I'd be interested to see what you come up with. Good luck!