Try looking up the central units model number. If you can find the unit on the web, might find some info on the thermostat. These companies make good profits on replacement parts, so you should be able to find what you need there. Now my AC and Heating was a custom install (what we could piece together, cheap). Only had to replace a start cap on the outside fan motor (****ing neighbor kids). It's been great for 11 years.
Home Depot sells remote control thermostats, but how often do you adjust it? A normal thermostat goes for around $30. Don't remember exactly, but think there are 4 wires, look like telephone wire, and these control the inside and outside units. I wouldn't think it too important which type thermostat you use.
Making your on replacement remote, for which you might use only a couple of times a year... Think it would be quick, simpler, and cheaper to just replace the thermostat.