I find the mechanical idea most appealing.
I’m imagining something like a barrel configuration with 3 faces that spin around a horizontal axis, with a one-way spring powered gear drive to rotate itself when the top face is depressed. Maybe that doesn’t exist yet. I’ll begin my search.
Or how about LED’s in the surface with a hand-crank generator disguised as a lazy susan underneath the board. Just spin it back and forth during analysis and it keeps the capacitor charged
I like the idea of levitating stones, but suspending marbles with air jets seems to be quite challenging and maybe impractical from an engineering perspective.
How about making a net grid using very thin filaments (like fishing line) pulled taut across a frame? That could give the illusion of a nearly invisible board.
Closer to a more traditional aesthetic would be to simply draw the grid lines onto a glass table top. I would not be surprised if that has already been done.
Now I’m thinking about thumb wheels like in a combination lock, with three positions to indicate white black or empty at each space
That seems very doable (and reasonably toddler-safe, to boot).
And it would work on the Space Station
For buttons, the mechanism seems like it’d be similar to how two-position pens work, but instead of a notched rod for a two position pen, you’d have a three-notched wheel moded in to the side of the button pushing against the ratchet pin. The pin/ratchet would partially move the button, after which you’d have a small triangular extrusion held in a slot where the flat face of the triangle would orient the upward face properly and stop it from turning if the button wasn’t depressed. If you didn’t want to have to push the button into the board, you could add a clear plastic cap on the outside, or else curve the faces upwards so the middle of the face would remain closer to the surface of the board while you depressed it to engage the mechanism.
I drew a picture of it, but I don’t have a phone capable of taking a picture of that picture and sharing it here. If anyone actually wants to build this, I’d happily do a 3D model of it 
Well, you could redraw it using MyPaint, Inkscape, GIMP, MS Paint,…
or, perhaps more appropriately, FreeCAD
How about making each button/“stone” square-shaped? Why not, if we’re considering new board aesthetics.
The button would be clear plastic, with the rotating triangular prism inside. Making the button square would make it easier to accommodate the rotation while still filling as much of the space visually as possible.
I think the I understand what you are suggesting without the picture. The triangular prism is in contact or close to the clear button casing in normal position. Pressing down on the button moves both the clear casing and prism down together, but as they come back up the prism lags behind a bit in order to rotate before coming back in contact with the casing.
I was initially thinking that the buttons would be square, but with images of an empty intersection, an intersection with a white stone, and an intersection with a black stone. It probably wouldn’t be too difficult to do circular buttons, though, but they’d have to be shaped a bit more. Square buttons would be closer to level with the board.
If the action on the buttons was relatively short, it might look better without the clear plastic domes.
At playgrounds, you sometimes see a tic tac toe game built like this:
I guess that same concept could be scaled down to make an all-in-one go board. Some sort of mechanism could be added to the axle to force the rotation “snap” into one of the three states.
I guess a downside of this general concept is that something else (e.g., memory, pen-and-paper, tokens, sliders, dice?) would be required to track the number of prisoners (if one wishes to use a rule set that counts prisoners).
Well prisoner counters would be trivial - just add Skurj’s prosed ‘combination lock’ close to the edges.
You could even hide them in a fold-out thing fanning out from the sides.
Of course, there are many ways to implement a counter. However, it is another design and aesthetic decision to consider. While one person might like that convenience, another might find the additional complexity (even if hide-able) unpleasing.
A combination lock style counter seems a bit inelegant to me. Personally, I would much rather just play by Chinese rules to avoid that and for other reasons.
RE: playing go in space
Two astronauts Koichi Wakata and Daniel Barry played go in space in 1996. See page 7 of this document:
Note that they mention that the board/stones are simply made of paper, due to the primary design concerns being mass and bulk. Also, any electronics are avoided due to the potential risks.
Here is a photo:
This thread was mildly interesting, right?
Yeah I’m a little disappointed that I have yet to implement any of these ideas
Two years later. Did you …?
An easy way to create a go board would be use clay and press the marbles a little bit into the clay (to avoid them from rolling away).
With clay pigments and glaze you can create a unique board.
And since ceramics is probably the oldest art form, I suspect there will be a pottery close by.