Make the blackberry syrup: In a small saucepan, combine the blackberries, sugar, and 1 cup of water. Bring to a gentle simmer over medium heat, stirring occasionally, until the sugar dissolves and the berries break down, about 5 minutes. Mash the berries with a fork or the back of a spoon to release their juices. Remove from heat and strain the mixture through a fine-mesh sieve into a bowl, pressing out as much liquid as possible. Discard the solids and let the syrup cool slightly.
Juice the lemons: While the syrup cools, roll the lemons on the counter to loosen the juices. Cut them in half and squeeze out the juice, straining out any seeds and pulp.
Mix the lemonade: In a large pitcher, combine the freshly squeezed lemon juice, the blackberry syrup, and the remaining 3 cups of cold water. Stir well to combine.
Serve: Fill glasses with ice and pour in the lemonade. Garnish with fresh blackberries and lemon slices, if desired. Enjoy!