For the mint ice cubes: Fill 2 ice cube trays with water. Distribute the mint leaves and zest among the compartments. Freeze until solid.
For the blackberry simple syrup: Put the blackberries in a medium bowl, reserving a few for garnish. Combine the sugar with 1 cup water in a small saucepan. Cook over medium-low heat, stirring occasionally, until the sugar is dissolved. Pour the simple syrup over the blackberries in the bowl and let steep for 20 minutes. Strain and let cool completely.
For the honey water: Combine the honey with 1 cup warm water in a small bowl.
For the cocktail: Combine the lemon juice, bourbon, 1/4 cup blackberry simple syrup and 1 tablespoon honey water in a cocktail shaker filled with ice. Shake and then strain into a glass filled with some mint ice cubes. Top off with the sparkling water and garnish with the blackberries. Reserve the remaining mint ice cubes, blackberry simple syrup and honey water for additional drinks.