1. Cook blueberries, water, and 1/4 cup sugar in a 1- to 1 1/2-quart heavy saucepan at a full boil, stirring, 1 minute. Transfer to a blender and blend until smooth. (Use caution when blending hot liquids.) Force through a fine-mesh sieve into a bowl, pressing hard on solids, then discarding them. Set bowl in a larger bowl of ice and cold water and let stand, stirring occasionally, until cold, about 5 minutes.
2. Fill a large cocktail shaker three-fourths full of ice cubes. Add blueberry syrup, rum, and lemon juice and shake vigorously 5 seconds. Add sugar to taste, shaking to combine, and strain into 4 (12-ounce) glasses filled with crushed ice.