Preheat the oven to 180C (350F). Lightly butter a 22cm (8 inch) round pan or use a corrugated paper baking round.
For the streusel:
Whisk together the sugar, flour, lemon zest and salt in a medium-size bowl. Mix in the softened butter into the flour mixture until it resembles granola, some larger chunks, some smaller. Set the mixture aside.
For the cake:
Cream the butter and sugar until light colored and fluffy. Add the egg and beat until combined. Be sure to scrape down the bowl. Beat in the yogurt and vanilla.
In a separate bowl, whisk together the flour, baking powder, baking soda, salt and lemon zest. Add to the batter, mixing only until combined. Do not over mix or the cake will be tough. Gently fold in the blackberries until well distributed throughout the batter. Pour into the prepared pan and spread evenly. Sprinkle with the lemon streusel mixture until the batter is completely covered.
Bake for 30-35 minutes or until a toothpick inserted into the center of the cake comes out clean. Cool on a rack for 15-20 minutes before serving.