I might not bother finishing this one, but figured I'd post the pics to put the info out there for other people.
I don't think I'll use the thing after it's done, so I'm not sure it's worth buying the parts to finish it or the time it will take to configure it.
This started with a DreamGear Mini Retro Arcade Machine that I picked up on clearance at Target for $15.
I bought it because it was cute and included 300 games. The games all suck pretty bad though. A lot of them are limited to one static playfield.
The controls feel good though, much to my surprise.

Disassembly:
Remove screws under battery cover.

Cut these foam strips across the seam underneath them.

The top is held in place by two tabs. Either free up the bottom and lift upward, or pry the top panel upward.

The bottom panel is held in place by melted tabs. They could be ground down, but It doesn't really need to be removed.

On the back panel: On/Off switch, speaker, battery compartment
In the cab: Main PCB is mounted behind the screen, a ribbon cable goes to the CP which has it's own PCB.
The CP PCB is just buttons and switches though, so the ribbon cable can just be hacked off and attached to a different interface.

The two things sticking down are LEDs to illuminate the little logo on the front.

The PCB must be used in multiple devices. It looks like it might fit in the Gameboy knockoff DreamGear makes.

The CP slides into a slot from the back. It has tabs on it that extend half way up the back of the monitor bezel.
These tabs have adhesive on them gluing them to the back of the monitor bezel, but they come apart without damage.

The screen is 2.5" I do not see a way to use it with the Pi Zero. I do not know much about driving screens though.

The joystick has arms underneath the CP that activate pads similar to a gamepad.

This board looks fairly labor intensive for the manufacturer to build.
I don't see anything on here that means anything to me, but maybe it will to someone else.

Ready for new guts:

I don't think a full size Pi would fit, but the Pi Zero wouldn't be a problem.

Reasons I'll probably shelve the project for the time being:
1. The Pi Zero outputs digital audio through HDMI, but doesn't output analog audio for a headphone like the regular Pi. Building the audio circuit is simple, but it takes up room and then it has to be configured on the Pi. There is a guy selling a tiny board on ebay that uses surface mounted parts. Even if I had that, I might still need a little audio amplifier which will also need to be powered. The charging/power board I already bought can supply an amp if needed.
2. I don't see a way to use the original screen and 2.5" screens with AV input are more expensive than I expected (around $45). Most of the Pi Gameboy type builds use the pins on the pi to interface the screen, but that bypasses the GPU (from what I understand). I'd get more bank for my buck using the AV connection.
3. I have no experience with the Pi or Linux. I was hoping I could download an SD card image, then plug 'n play. The audio will have to be set up though. I also don't want any emulators other than MAME, so I don't want them showing up in the menu. I already have the Pi, so I will experiment and see what I'm getting into.
Mostly I hate to keep sinking money into something that I know I won't use for more than a few minutes after it's completed.
I'll probably have $100-125 in this thing by the time it's finished. I could build a real cab (using parts I have laying around) for that much.
I could use some of these components on other projects that I'll actually get more use out of.