Make berry filling. Mash blackberries in a saucepan until broken down. Strain through fine mesh strainer, pressing berries to extract about ¾ cup juice. Return juice to clean pan. Add the lemon. juice. Whisk sugar and cornstarch together, then whisk into juice off the heat. Bring to a boil over medium-high heat, stirring constantly until thickened. Pour into bowl to cool completely.
Prepare crust. Preheat oven to 350°F. Line 9x13 pan with foil or parchment, leaving overhang. Mix graham cracker crumbs, sugar, melted butter, and salt until moistened. Press into pan bottom. Bake 8-10 minutes until lightly golden. Cool on wire rack.
Make cheesecake filling. Beat cream cheese until smooth and creamy. Add sugar, flour, sour cream, and lemon zest; mix until combined. Add beaten eggs and vanilla, mixing just until incorporated.
Assemble and bake. Pour cheesecake filling over cooled crust. Drop 10-12 tablespoon-sized dollops of cooled berry filling on top. Swirl gently with knife to create marble pattern. Bake 30-35 minutes until edges are set and center has slight jiggle.
Cool and chill. Cool at room temperature for 1 hour, then refrigerate at least 2 hours before cutting. Use foil overhang to lift from pan before slicing.