Bring 1 1/2 cups water to a simmer in a medium saucepan. Remove from the heat, add the tea bags and let steep 5 minutes. Remove the tea bags (do not squeeze). Let the tea cool to room temperature.
Meanwhile, bring 3 tablespoons water, the sugar and ginger to a simmer in a small saucepan over medium heat; cook, stirring, until the sugar dissolves, about 2 minutes. Remove from the heat and let cool; discard the ginger. Stir the ginger syrup into the tea. Pour into ice cube trays and freeze until solid, 2 to 4 hours. Freeze the chopped melon until solid, at least 2 hours.
Combine half each of the tea ice cubes, frozen melon, ginger ale, lemon juice and mint in a blender and puree until smooth. Divide between 2 glasses. Repeat to make 2 more drinks.