I have seen such a problem very recently with one of my products. The problem seems to be(at least till now) that the LCD bezel tabs (the ones that hold the LCD together with its PCB) were in close proximity(around 0.5mm closest point) of the relays that were switching on/off some motors in my case. Just like you experienced, I would get the same behaviour even when the motor was not powered by closing the electrical circuit. Just one wire would drive the lcd mad. What I did was to bend away the offending tabs on the LCD so as to increase the air gap beyond 2mm. Now, the problem has not totally died away, but the frequency of occurence has gone down manifold.

My suggestion is that you look for any possibilities that the circuit is getting coupled due to HV breakdown of air. Try to increase distances between the HV and LV sections to whatever may be practical. You might want to consider occasionally re-initializing the LCD if it still troubles you.