Put the blueberries, mint leaves, sugar, and lime juice in the bottom of a tall glass.
Use a muddle or the end of a spoon and crush the ingredients together, including bursting most of the berries.
Fill the glass with ice.
Pour the rum over the ice.
Top off the glass with club soda. Garnish with more limes, mint, and blueberries as desired.
Turn this drink into a mocktail (without liquor) by omitting the rum and replacing with more club soda, tonic, or lemon lime soda.You can use a sweeter mix in if preferred (instead of club soda) such as tonic or lemon lime soda.I do not recommend using simple syrup in place of the sugar unless you want a very sweet drink.These can be made in a batch in advance by combining all ingredients except the club soda together. Top off each glass as poured with the club soda.