Muddle blackberries and ginger honey simple syrup in a cocktail shaker. Add remaining ingredients and shake for 20 seconds. Add ice and shake until the cocktail shaker is chilled on the outside. Double strain into a glass with ice.
Honey Ginger Simple Syrup
Combine all ingredients in a small saucepan and bring to boil. Reduce heat and simmer for 5 minutes. Cool for 10 minutes. Strain into a jar. Store in the refrigerator for up to 2 weeks.