Heat the water, sugar and mint leaves in a small saucepan over medium heat. Stir continuously until the sugar dissolves.
Remove from heat, pour in a small bowl and refrigerate the sugar-mint mixture for 30 minutes.
Once chilled, pour the sugar mint syrup through a strainer (to catch the mint leaves) and into a large pitcher.
Add the rum and lime juice. Stir to mix well.
Place 4-5 blueberries into the bottom of each mold. Pour the mixture into the molds, filling halfway. Add a mint leaf to each ice pop mold along with 6-8 more blueberries. Top off with additional liquid, up to the fill line.
Insert the lids/sticks onto the molds and press down firmly. Freeze overnight.
Before serving, run warm water along the bottom of the molds to gently release the ice pops.