The broken electronic thermometer probably used a thermistor probe. You need to measure resistance at three known temperatures to calculate the characteristics of your particular thermistor.
To read a variable resistor you can't just hook it between A0 and a power source. You also need a fixed resistor for A0 to Ground. The variable resistor and fixed resistor act as a voltage divider. A0 then reads the resulting voltage and some math lets you calculate the unknown resistance.