I'm going to assume you are new to this, so If I explain too much, don't be insulted... just trying to help.
The only thing that's not super obvious is what a solinoid is... it's just a big fat relay, which is just a switch you control with electricity to turn it on and off.. You use a skinny wire to to turn the switch on and off, a fat wire is what gets controlled by the switch. The solinoid on the FJ is a little cylinder built onto the starter.
You use a skinny wire to control a fat wire, thats all a solinoid does.
There are basically 2 separate parts to the starter battery circuit ,
Part #1 the contol circuit (skinny wire)
battery + ---->key switch -------> Starter Solinoid Control (turns starter on and off using low power---> ground
This is a low power circuit, using thin wire that just toggles the solenoid on for the starter, its part of your wiring harness
The power from your switch, is a low power signal that controls the solinoid.
Part #2 The high power signal is usually from the battery to the big lug on the starter. (fat wire)
Battery+====> Starter solenoid Power===> starter Motor ====> ground
Ignore that is diagram is 24v and take a look.
Blue is your fat wire high power circuit.
Green is your low power thin wire circuit. it shared the ground through the body of the starter with the blue circuit.
The first question is is low voltage your mechanic identified to the control circuit (green) or the power circuit (blue)
The thing is, when most people jump a starter they go straight from the battery to the power side, to test, not the control side.
If it's the control circuit, then replacing the switch -MIGHT- work, but it's probably a bad ground or corrosion somwhere in the control circuit, which the swith will only help a little at best. (Otoh, its really easy to swap)
On old toyotas I always start by assuming it's the ground. It's a pain, but you can totally address this yourself. Start by cleaning up all the grounds on the control circuit..
My guess is it's the power circuit. Luckily thats stupid easy to upgrade so it works better. 4 bolts and 2 thick ass wires. that's it. You can 100% do this your self, if you can turn a few bolts and operate a tape measure.
I had the same problem with the starter on my Fj40, I added a new big fat ground wire (2awg or thicker, marine grade wire is better) from a mounting bolt on the starter right to my battery -, and it went away. I didn't even remove the old wire , just ran a new one in parellel.
If you also add a big fat wire from your battery + to the + lug (Big) on the power side of your starter solinoid, that would be even better, I need to do this on my 40.
I suggest 2awg or thicker, I'd probaly go 2/0 (double zero) and just be done with it , You just need to measure how long a wire you need and order it on amazon.
Here are the parts of the starter, double check I got the big and little wire right, I'm guessing based on the size of the bolt. just look at the wire going to it.