Preheat the oven to 325°F (160°C).

In a mixing bowl, combine the graham cracker crumbs, melted butter, and ¼ cup sugar. Press the mixture firmly into the bottom of the springform pan to create an even crust. Set aside.

In another mixing bowl, beat the softened cream cheese with an electric mixer until smooth and creamy. Gradually add 1 cup sugar, sour cream, eggs, vanilla extract, and lemon juice, mixing until fully combined and smooth.

In a saucepan, combine the blackberries, ¼ cup sugar, lemon juice, cornstarch, and water. Cook over medium heat, stirring gently until the blackberries break down and the mixture thickens, about 5-7 minutes. Remove from heat and let cool slightly.

Pour half of the cheesecake filling over the prepared crust. Swirl in half of the blackberry mixture. Add the remaining cheesecake filling, and then dollop the rest of the blackberry mixture on top. Use a knife to swirl gently for a marbled effect.

Bake in the preheated oven for 55-60 minutes until the center is set but slightly jiggly. Turn off the oven and let the cheesecake cool in the oven with the door slightly open for 1 hour.

After cooling, refrigerate the cheesecake for at least 4 hours, preferably overnight, before removing it from the pan and serving.