OK, first the terminology.
What you are referring to as strands are called conductors. You have solid conductors and you have stranded conductors. Solid conductors are a single piece of wire. Stranded conductors are many small wires twisted together to make up a wire the equivalent size of a single solid wire. Stranded wire is used in things like lamp cords. Stranded wires is much more flexible than solid wires.
Next you have cables. A wire is a single wire, a cable is two or more insulated wires or conductors in an outer covering. Outer covering is called sheathing,
There are two ways to wire a light fixture. First is called feeding the switch. When feeding the switch the incoming power cable is taken to the switch box. The neutral (white) is connected to the neutral going to the light fixture. The incoming hot is connected to one terminal of the switch. The hot going to the light is connected to the other terminal of the switch. When switch is on current passes through the switch, goes to the light, passes through the light and returns on the neutral.
With this wiring method you only have one 2 conductor (3 wire) cable at the light box unless you have additional lights on this switch. I say 2 conductor cable because you don't count the ground, it is not a conductor and really not part of the circuitry.
The other method of wiring a light fixture is called feeding the fixture. Using this method the power in cable is taken to the light fixture box. The neutral is connected directly to the light fixture. The incoming hot is connected to one of the conductors of another 2 conductor cable that goes to the switch box. So the current comes in on the hot, goes to the switch on one conductor, goes through the switch and comes back on the other conductor. That return conductor is connected to the black of the light fixture. That cable to the switch is called a switch loop.
It has long been the practice, and now is the code, that the white will be used to take power to the switch and the black will be used to return. That gives you a black to black connection at the light fixture.
You are also required to designate the white as a hot conductor by coloring both ends some color other than white or grey. Usually black electricians tape is used. However, this is frequently not not done, especially by the pros.
So if you have two 2-conductor cables at the light box and do not have any other lights on this switch you have a switch loop.
However you said,
I tried connecting the 2 neutrals from the switch side to one of the neutrals from the GFI side
I assume that means you connected 2 neutrals (white) of the switch circuit to the neutral of the GFI circuit. To have 2 neutrals at the switch box implies that you are feeding the switch. Meaning that the cable to the light is not a switch loop.
So, do you have multiple lights on this switch.
Also this is not a 3-way switch is it (how many terminals does it have)?
Finally, if connecting the neutrals of the switch circuit to the neutral of the GFI circuit made the light work, that implies that my original logic was incorrect.
You could have a poor connection on the neutral side just as described on the hot side. One where you have a connection but is so bad that it limits the amperage flow. That would give you the same voltage readings you previously described. I guess I should have considered that possibility also.
Connecting the neutrals of the switch circuit to the neutral of the GFI circuit could have caused the GFI to trip, especially if they are on different halves of the cycle.
So, do you have additional lights on this switch, how many terminals does the switch have, did the GFI trip? If you have additional lights do they work properly?
I recognize that this is probably more than you ever wanted to know.