The inventor of the mercury thermometer is Gabriel Fahrenheit, who built it in 1714. The inventor of the first thermometer is Santorio Santorio, who built it in 1612.
Many people have claimed that Galileo is the inventor of the thermometer, but actually he invented a thermoscope and not a thermometer. A thermometer measures the changes in temperature while a thermoscope simply shows that there are changes in the temperature.
Fahrenheit's invention of the mercury thermometer improved upon Santorio's thermometer with its more accurate measurements and advanced glasswork. It was Gabriel Fahrenheit who designed the first standard thermometer scale as well, which is how the name "Fahrenheit" was chosen.