In a small saucepan, place the blackberries, sugar, and water and cook at medium heat for about 10 minutes. Crush the blackberries while they are cooking and stir occasionally.
Remove from heat and pass mixture through a strainer. Let the blackberry mixture cool at room temperature. Reserve.
In a small pitcher, combine ginger beer, vodka, lime juice, and blackberry syrup. Stir to combine.
Add ice to two copper mugs and add the cocktail mixture. Garnish with mint and fresh blackberries.