To make simple syrup, bring the water to a boil. Dissolve the sugar in the boiling water, stirring constantly.
Once the sugar is dissolved completely, reduce the heat, cover, and allow to simmer for 10 to 15 minutes (the more sugar, the less simmering time).
Remove the pan from the heat. Allow to cool completely and thicken, then transfer to a glass bottle or jar and store in the refrigerator (will last refrigerated up to 1 month).
To make cocktail (you can make multiple cocktails at once), add blackberries, mint, lime juice, squeezed lime half and honey or simple syrup to a cocktail shaker or pitcher. Muddle all of the ingredients together to coarsely crush the fruits and mint leaves. Remove lime, add vodka and shake or stir well. Serve into prepared glass (not straining) and slowly add the bubbles. Garnish drink with a fresh blackberry and a lime wedge. Drink and enjoy!