Step 1: Macerate the Fruit - Combine the blackberries, sugar, lemon zest, and mint leaves in a mixing bowl. Using a potato masher or sturdy spoon, gently crush the berries until they've released much of their juice and everything is well combined.
Cover the bowl and refrigerate for 2 days. During this time, the sugar will draw the juices from the berries, creating a rich, flavorful syrup.
Step 2: Strain the Syrup - After two days, pour the mixture through a fine mesh strainer into a clean bowl. Press firmly on the berries with the back of a spoon to extract as much juice as possible. If you notice sugar still clinging to the berry solids, strain the liquid through again until most of the sugar has dissolved into the syrup (it may take a few times).
Step 3: Add the Vinegar - Whisk the apple cider vinegar into the strained berry syrup until fully combined. Using a funnel, pour the finished mixture into a clean quart jar and secure it with a lid.
Refrigerate for about one week to allow the flavors to mellow and develop before serving.