The light will come on if temp goes over the top (me thinks 120) or coolant level is low.
If it comes when engine is cold and coolant level is between min and max try to add more coolant, so it is just a bit over max. This should cure the red light, if not try to clean the low coolant sensor (electrodes) with an old toothbrush - should this not help check wiring to expansion tank or possible expansion tank swap.
Also worth measuring the coolant temp sender (posted an image here, just look in a recent temperature related thread)... but if this went wrong you'd probably notice since the temp gauge would show high temp...