Our furnace pilot goes out after the unit has not been used for any period. It stays on once you relight the pilot and then randomly will go out over the next couple of days, after that initial period it stays on consistently.

We have already changed the thermocouple, cleaned the unit, and checked for drafts. We suspect the culprit might some water coming from a part when the unit gets cool, I think the part is labeled an air something or other - can check it out when I am home.

Any thoughts on what this might be and how to fix it?

I`m curious to learn what a air something or other is
If it has a pilot light its likely a 80% furnace, so it doesn`t make water,so it is leaking water from somewhere when it rains or ac coil condensation pan could be leaking, if your water theory is correct
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Did you remove and clean the pilot light assembly,sounds like you may be a little stopped up and not getting much gas pressure for pilot light