In a medium saucepan over medium-high heat, combine the sugar and 1/2 cup 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. Let the syrup cool completely.
In a blender, combine the berry syrup, rum and lime juice. Add the ice and blend until smooth. Serve immediately.