In a medium saucepan over medium-high heat, combine the sugar and 1/2 cup (120 ml) water and bring to a low boil, stirring until the sugar dissolves. Add the blueberries and reduce the heat to a simmer. Use a wooden spoon to crush the berries, allowing their juices to release, and continue to simmer for 20 minutes. Press the berry mixture through a fine-mesh sieve set over a bowl and discard the solids.
In a blender, combine the berry syrup, rum, and lime juice. Add the ice and blend until smooth. Serve immediately.
Recipe courtesy of Katie Lee, Endless Summer Cookbook (Stewart, Tabori & Chang / ABRAMS , May 2015)