I would call it either a control panel or a button panel, but would defer to whatever the owner's manual calls it in a schematic.
bizarrely, those thermostat controllers can be pretty cheap to replace and totally in the realm of DIY.
the wires leading to HVAC should be color coded and sometimes are tagged.
just saying if trying to fix a cheap component becomes too infuriating (because they are often cheaply made).
anyway, good luck