What triggers the rain?

It appears that the rain occurs on a timer that you can view on your map. See this article:

Bring up your map. The little dots around the rung / season meter in the bottom left of the screen counts down to when it will start raining. Keep an eye on it.

I'm not sure exactly how long this timer lasts, but it seems like it is pretty consistent. Based on articles like this one, it seems that the rain comes at a regular interval and doesn't appear to be affected by any other factors.

This comes periodically. I never timed it, but it felt like maybe every fifteen minutes or so.

As an aside, I found articles mentioning the rain coming after "spending too long without hibernating," like this IGN review, but I believe the author was referring to the fact that you're required to hibernate on a regular interval lest you die to the rain.