To a small pot on the stove, add 1/4 cup of honey plus 1/4 cup of water and heat until the honey dissolves. You only need 1 tablespoon of this for 1 serving, so reserve the rest in the fridge, it'll keep for up to 5 days. Set aside to cool.
Add the mint leaves, blueberries and lime juice to the bottom of a glass.
Use a wooden spoon or cocktail muddler to mash, or muddle, the blueberries and mint leaves for about 30 seconds. (see notes if you prefer a smooth texture with no blueberry/mint pieces)
Add the simple syrup once cooled, plus the rum stir to combine.
Top with ice and fill the rest of the glass with sparkling water, club soda or seltzer.
Serve with extra blueberries, lime wedges and mint leaves if desired. Enjoy!