In a small saucepan, combine sugar, water, and 25 mint leaves. Heat over medium heat until the sugar is completely dissolved, then remove from heat and allow to cool to room temperature.
Once cooled, strain out mint and pour syrup into a large bowl. Squeeze into the bowl lime juice and rum and mix until combined.
Pour into 10 popsicle molds, filling a little more than ¾ of the way up. Toss one fresh mint leaf and about 12 blueberries into each mold. Fit with popsicle stick and freeze for at least 4 hours, preferably overnight.