This could indicate a shock hazard, so do be careful while you're diagnosing the problem. In particular, if the motor windings were leaking current to ground (the motor casing and by extension the saw itself), a properly functioning GFCI could exhibit this behavior. Especially avoid touching a metal surface saw while some other part of your body is grounded. Safer yet, avoid using the saw until you have this figured out.
You would need other tests with test equipment to tell, but trying with another GFCI might be an accessible option.
One possible scenario is that the motor windings heat up under load and cracks or other weak spots in the insulation then allow current to flow to ground without passing through the Neutral wire.
Depending on the motor type, it could also be leakage in the run capacitor, an insulation breakdown in the switch, or in any part of the wiring (especially suspect given its age), but a systematic diagnosis here is the best bet. Such a diagnosis would require understanding the electrical principles involved and may be best left to an expert.
I suppose you could eliminate certain whole sections of components by, say, wiring the motor directly to a plug. If that fixed it, the switch and wiring would be suspect; if it still happened, it narrows it down to the motor.