are you sure your probe(s) aren't just fried from taking on water or being over heated? The erratic temp graph you posted looks like what I got with just about every probe I've killed one way or another.
As for troubleshooting the HM board to solve this problem, there's not a whole lot to it. The probes come in through the input jacks, there are only 3 components between the probe jack and the ATMega, those are the sets of three components that stand on end behind the probe jacks (the resistor/capacitor from the RC filter and the pullup resistor). After the RC filter and pullup resistor the probe leads go directly to the ATMega pins, starting in the corner down behind the pit probe jack, working back from the corner Probe-0 (PIT) through Probe-3.
So you can reflow solder on the components that are standing on end, and the four pins of the ATMega in that corner, and the probe jacks. Then clean that area of the board with some isopropyl alcohol and a soft bristle brush (like an old toothbrush). It could also be erratic 3.3v supply to the pullup resistors causing the bad readings, so measure that to make sure it is stable.
If you still have the noise when running off a battery then you can rule out power supply interference... My first guess would be bad probes...