Preheat your oven to 350°F (175°C) and line the muffin tin with paper liners or grease it lightly with butter.
In a large mixing bowl, whisk together the flour, baking soda, salt, and bran until well combined.
In another bowl, combine the melted butter, light brown sugar, and molasses. Mix until smooth, then add the shredded zucchini and sour cream. Stir until everything is well incorporated.
Pour the wet mixture into the dry ingredients. Gently fold until just combined; be careful not to overmix, as this can lead to tough muffins.
Gently fold in the blackberries, ensuring they are evenly distributed throughout the batter.
Scoop the batter into the prepared muffin tin, filling each cup about 2/3 full to allow for rising.
Place the muffin tin in the preheated oven and bake for 20-25 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, remove the muffins from the oven and let them cool in the tin for a few minutes before transferring them to a wire rack to cool completely.