Remove skin from honeydew and cut melon into chunks. Blend until a smooth purée forms.
Strain purée through a fine-mesh strainer until all pulp is removed. Measure honeydew juice, then weigh an equal amount of sugar.
Combine juice and sugar and stir until sugar dissolves. Warm mixture over low heat, being careful not to boil.
Add tarragon sprigs and cook for 30 minutes or longer, stirring occasionally, until syrup is lightly infused with tarragon flavor and aroma. The syrup will take on a light golden-brown color as it infuses.
Remove sprigs and refrigerate syrup until ready to use.