If your moist weather is causing condensation to form on some areas that already have smoke residue, that moist residue will easily conduct enough electricity to cause what everyone is referring to as "leakage".
GFCIs operate by comparing the current in the hot leg with the current in the neutral leg. If the two are not in perfect balance, you know that some current is finding an alternate path to ground (not taking the neutral wire).
Smoke residue is often slightly conductive. Add some moisture, and it can be very conductive.
It is easy to end up with residue or buildup of smoke tar in the electrical areas of a smoker.
If there is a "bridge" of smoke residue that can "leak" current from the hot side of things to ground (bypassing the path through the neutral wire), this will trip a GGCI.
What is probably happening is that you have smoke residue somewhere that allows this "leakage" to occur, but only when it is moist. As the smoker heats up, it drives the moisture out of the smoke residue, and the conductivity goes down, reducing the amount of current leakage, and allowing the smoker to run without tripping the GFCI. But as soon as you have the right conditions to allow moisture to condense into the smoke residue, the conductivity goes up again to a high enough value to trip the GFCI.
Finding the location(s) where this "bridging" is occurring will be largely a matter of visually inspecting suspect areas. Look for places where there is a "hot" connection that is not insulated and has a likely path to a grounded surface. If there's some residue built up, it will need to be cleaned away. Do this with the smoker unplugged, of course!
I'd use a solvent that dissolves grease to help with this. Something like hexane (zippo lighter fluid) would work well, but it's flammable as heck! Some electrical contact cleaners are nothing but a solvent, and could be good. A toothbrush can be great for scrubbing (with solvent) such areas.
This may be somewhere well hidden. The smoke can get into various places easily, but as was suggested above, check right near the connections to the heating element as a start.
This might be really easy to find and fix. Or it might be hidden pretty well.