One possibility is moisture getting inside the element. It sounds similar to a problem I have had. I use a rebuilt drop-in electric range top as my brew stove (I rebuilt it to get three heating coils close enough together to get my 10 gallon Megapot over them.) During initial testing, I found that the GFCI would trip after a minute or so of heating, and any one of the three coils alone would trip it. If I turned the stove off for a few minutes, and then back on, it would trip more quickly, sometimes immediately, sometimes after only a few seconds. I wanted to do a test to see how long it would take to heat my water to boiling, so I temporarily wired around the GFCI to run the test. The day after the test (45 minutes with the stove on full blast), I put the GFCI back in the circuit to trouble shoot the problem and the problem was gone. My conclusion is that the coils get moisture inside them, which vaporizes after a minute or so of heating, causing the GFCI to trip. If you continue heating the coil (without the GFCI in the circuit) it bakes the moisture out and everything is fine for a while, but the coils will accumulate moisture if left unheated for weeks or months. I solved my problem by simply wiring a regular 30 amp breaker inside my GFCI box, in parallel with the GFCI breaker. At the start of the brew day, if the GFCI trips, I turn on the regular breaker, run the stove for 5 minutes to bake out the moisture, then turn off the regular breaker and turn the GFCI breaker back on and continue without problem.
I know those water heater elements are designed to be used underwater, unlike the coils on my stove, but it does sound to me like water could be getting either into the element or around the connections to the element, and it wouldn't take more than a very tiny amount to trip a GFCI.