So, I went back to my pencil and paper and sketched out a completely different idea using what are called edge connectors. This connector type would be mounted vertically on a base board and each LED board would plug into a connector. In effect, the LED boards would become "verticals" and the number of unique circuit boards would reduce to two -- the LED board + the base board.
A few years ago the folks at Looking Glass created an 8x8x8 cube kit, the L3D, using a technique like this. They had a decent Kickstarter campaign for it: https://www.kickstarter.com/projects/lookingglass/l3d-cube-the-3d-led-cube-from-the-future but it appears they no longer produce this item.
The way this works is by etching signal pads on the LED boards directly -- no separate connector is need on the LED boards. These signal pads would then make contact with the edge connector pins when the LED board was inserted into the edge connector. The total number of connectors would reduce from 256 down to just 64.
And it would be simpler to assemble.
Then I looked at the cost, searching on Digi-Key for "Card Edge Connectors." For this type of connector I would need 4 pins: +5V, GND, serial data in, serial data out. Edge connectors connect to both sides of the PCB when the PCB is plugged in. Therefore, I would need two pads on the top of the PCB and two pads on the bottom of the PCB.
I narrowed my search to board thickness of 0.062", the most common PCB thickness and also the least expensive to manufacture. I found a nice little connector, but it only had one supplier -- that is a warning flag. What if you can't get parts when you need them? There isn't a second source. Add to this the fact that single piece pricing is $2.41 and 1000 piece pricing is $1.60. Even at 1000 piece pricing, 64 of these = $102.40.
Way. Too. Expensive.