What is the tendency of a solute to change the melting point of a solid called?
Usually it is called bloody annoying. Melting point depression is what it is formally called.
A substance that has some contaminant (which could of course include the recrystallizing solvent) usually has a melting point a little less than that of the pure substance. Given the fact that the preparation of a few solid derivatives of an unknown compound, and measurement of the melting points of its derivatives, is still the best way to identify an unknown, melting point depression can be a bit misleading.