Answer:
Either call for a service tech, or buy a new one. A mini fridge may cost less to buy new than to call for service, unlike a conventional refrigerator.
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Another option, consider going to the local appliance parts store and asking for a "hard start" unit (they sell for around $25). If you hear a "click" when the thermostat is moved below the current temp, but the "click" is not followed by the hum of the compressor, then it's either the starter or the compressor itself. If it's the starter, then the hard start unit will fix you right up.