Preheat oven to 350 degrees. Butter an 8-inch square pan; set aside.
For the topping, in a large bowl, stir together the flour, sugar, and cinnamon. Using a pastry cutter or your finger, cut in the butter until the mixture resembles coarse crumbs. Refrigerate until ready to use.
For the cake; in a medium bowl, whisk together the flour, baking powder, ginger and salt. In a large mixing bowl, cream the butter with an electric mixer on medium speed. Add the sugar; beat until light and fluffy. Beat in the egg and vanilla. Alternately beat in the flour mixture and the milk until just combined.
Spread the batter evenly in the prepared pan. Scatter the blackberries over the batter. Sprinkle with the streusel topping. Bake for 45 to 60 minuets or until a toothpick inserted in the center of the cake comes out clean. Transfer to wire rack to cool slightly before serving.