Heat the blackberries, sugar and water in a small saucepan over medium-high heat. Use a potato masher to smash the berries while they heat, continuing to cook until a thick consistency is reached, about 10 minutes. Set a fine mesh sieve over a small bowl and strain the blackberry mixture through, pressing out as much juice and pulp as possible. Refrigerate for at least 10 minutes.
For each drink, fill a cocktail shaker with 2 tablespoons of the blackberry puree, 2 tablespoons of the mint leaves, 2 ounces of gin, 1 tablespoon of lime juice and a handful of ice. Shake 5 or 6 times, or until the drink is ice cold. Strain into a tall glass, top with 6 ounces of ginger ale, and stir gently. Once all of the drinks are made, top each with a big scoop of vanilla ice cream.