In a medium saucepan over medium-low heat, bring 1 cup water, sugar, and ginger to a gentle simmer, stirring to dissolve sugar. Remove simple syrup from heat, cover, and steep for about 30 minutes.
Over a medium bowl, pour syrup through a fine-mesh sieve, pressing minced ginger against sieve with the back of a spoon to extract flavor. Discard ginger. Cover bowl with plastic wrap, and refrigerate until well chilled.
Evenly distribute peach slices among 8 tall glasses. For each glass, lightly press peaches with the back of a wooden spoon to release their juice. Pour about 2 tablespoons chilled syrup over peaches. (Store remaining syrup in refrigerator for up to 2 weeks.) Evenly divide the rum. Add ice cubes; then top with soda. Stir to distribute syrup, peach juice, and seltzer evenly. Garnish with 2 mint leaves.