I have too many things brewing right now to pursue any kind of a repro on these boards, and zero ability to do a housing for it.
But this partial schematic should give a good leg up for anybody else that may want to take it on.
Doesn't mean I won't keep an eye on things and chime in as I can help.
Oddball research projects like these are just too fun for me sometimes.
The IC's used for both locations U1 and U2 on these I'm about 99.9% positive is an LM3914. Was very commonly used for LED bar graphs in things like graphic equalizers, Vu meters, voltage meters, and things like that back in the 80's. As soon as I saw how this operated, I knew this was the likely culprit. If I'm not going to be making them myself, there's no reason not to share the info. The more I dug through building a schematic, the more it all lined up perfectly. These IC's are still readily available either in SMT or thru-hole and not expensive.
The other critical piece you'll need is called a 3-wire slip joint. This is the three-wire connection between the motor board and the LED board. Lots of places out there to get them, but I don't know how fast you need to spin, so be careful of the cheap 250rpm versions out there. The three wires coming through are LED voltage, Signal, and Ground, as best I can tell. Not something I've worked before but I did know the name at least.
Motor board I'm not going to really dig into very much as the photos should be a very good start to reverse engineer things. All the parts are clear and easy to see, and from the various photos so are the traces. Generic transistor, voltage regulator, and op-amp. The only tricky part will be nailing down the value on the pot and the speed of the motor.
The slip joint, along with the housing, are going to be your major factors on price.
I don't expect this to be a cheap repro by any means.
Can't quite figure out a couple components on the LED board from the photo for a schematic.
No idea at all where the signal line from Pin2 on the slip joint goes.
Also can't trace the lines from R1, R4 and R5 completely around U2.
The two IC's do have different circuits so can't just copy.
I'm also not 100% certain how R7-R10 and C2-C5 really are, so definitely verify those if possible.
Won't be able to know the resistor values without seeing the stripes (which may be visible from a low side angle).
Multimeter can't always get you values if you measure in-circuit.
I am confused over which version Laser War had in it.
Pausing the youtube video suggests the 20-LED version.
Some of the older promotional photos LOOK like the 80-LED, but that could just be ISO/Shutter settings being tricky.
However, that old Worthpoint link to the EBAY photos? That appears to be a third version completely.
I wonder if they may have made games with all three (known) versions of the Lyte.
There may not be a 'correct' version. Too little data points here.
Lyte Interim Schematic.pdf
LYTE 20LED (resized).jpg