Elantra HD (2006-10) :: Climate Control Turns Off Sporadically

I purchased an '07 Elantra used about six months ago. Ever since I bought it, an issue with the climate control would pop up every few weeks.

Here's the issue: I'll hear a "click" under the passenger glove compartment, then the climate control would turn off. By "turn off", I mean if I have it on heat, then the heat would turn off. If I have it on AC, then the AC would turn off (including the little green AC light) and I get hot air (not ambient air from outside, but *hot* air). It turns off... then it turns back on. Then maybe a few seconds later it will do it again. Sometimes it turns off for a few seconds, sometimes it turns off for less than a second... for example... AC is on, blowing cold air... *click* :hot air: *click* back to cold air... a few seconds later, same thing happens.

The clicks seem completely random, and not triggered by any events, such as going over bumps in the road.