Late edit, check out pcky remix of this, its awesome!
https://www.thingiverse.com/thing:3990991

After watching Makers Muse's video I decided, that I want to experiment with this too.
So I designed the box from scratch in OnShape.

The Design

I designed it parametric and and made the dimensions easily editable.
You can customize it yourself, IF YOU USE ONSHAPE:

• Hight
• Size of the buttons
• Tolerance between the cover and the bottom part

Also there are a few different versions, I added after printing it. It includes rounded corners, a higher bottom for more space and different tolerances.

Electronics

I used an cheap Arduino Pro Micro ATmega 32U4 , as these can act as a keyboard. (Links are in German)

Edit: seanrhinehart over on Reddit recommended using
"Teensy family of products, as they have integration into the Arduino IDE, and specific project types and examples for USB MIDI, keyboard, joystick and other HID applications. https://www.pjrc.com/store/teensy32.html"
So, depending on your intended use, these might be a better choice.

The arcade buttons I used are Sanwa OBSF-24, wich have such a great feeling and sound to them!

Programm Code

I did the best I could and made it customizable. You can edit the keys / combinations / commands that should be send to your PC. Writing sentences is also possible. For that see the Arduino Keyboard reference
Also the used pins can be changed. Its a rather simple piece of code, so check it out and edit it :D
I added the program code to the files here in thingiverse.

In the current state pins 2,3,4,5,6,7,8,9 and 15 are assigned to LEFT_ALT + 1 to 9 as I did not have and shortcuts assigned to this combination yet. (If you do not want to edit anything just upload this file with the arduino IDE)

UPDATE:

I now used my tolerance test to figure out the best tolerance for a press-fit and its 0mm to 0.05mm. So it seems the Prusa MK3 prints really precise. If you own a good 3DP, just use the smallest tolerance and it should fit well.

Edit / Commentary

• I now added a 4x4 version with 0.15mm and 0.05 tolerance. 0.15 seem to be to wide. I hope 0.05 fits well.
• As I only printed the 0.2 tolerance version, It would be great, if you could tell me, how well you print fits. 0.2 was too wide on my Prusa MK3. 0.15mm also seems to be too big, so maybe 0.05? Zero tolerance works best on my Prusa MK3. It should be a press fit :)
• Also I am currently reworking the software, now using the HID-Project-library by Nico

If you have questions, or even better constructive criticism, leave a comment or text me :)