In a collins glass or highball glass, add the blueberries, mint leaves and simple syrup.
Muddle together to release the oils of the mint leaves and the juices of the berries. (See note for an alternate method.)
Top with ice, filling the glass all the way.
Add the lime juice and rum. Top with club soda.
Garnish with a straw, lime wedge and a mint sprig.
Notes
When muddled, the berries do leave behind some pulp and seeds. If you prefer a clearer drink without texture, do this instead: Muddle the berries, mint and syrup in a cocktail shaker or mixing glass. Top with lime juice and rum. Then use a strainer to pour the mixture into a glass filled with ice. Top with club soda and garnish. If you don't have a muddler, you can use the blunt handle of a wooden spoon.