Is it the crystal / quartz?
It's unlikely the crystal. It's hard to see from your photos, but it looks like the micro-controller is powered from the battery through a CDS cell and then a capacitor. So I'd guess that the micro-controller is meant to go into a micro-power mode and survive from either the leakage current through the CDS cell, or the energy stored in the capacitor.

I'd try first replacing C1 with a new capacitor that can cope with low temperatures and is low leakage, and failing that check the CDS cell isn't looking burned up.

Others have said the board looks like it's not been cleaned, however I think that's probably a conformal coating to cope with operating in a condensing atmosphere.

It'll probably emit nasty fumes and you stab through it with a soldering iron, and you'll very likely want to put something back when you're done. Acrylic nail polish is cheap and works, or use a can of proper electronic conformal coating.