Add water, sugar and blackberries to a saucepan. Heat on medium, stirring to dissolve sugar. Bring to a boil, then remove from heat.
Mash the berries with a potato masher and let sit for 10 minutes to flavor the sugar water.
Strain with cheesecloth or a fine sieve into a clean jar.
Let cool completely then store in an air-tight container in the refrigerator.
Notes
I figured 1/2 ounces as a serving. A recipe will call for 1/2 - 1 ounce depending on what you are making. When finished cooking, you will end up with less than 1 cup as the water evaporates and maybe slightly different each time you make it.
The flavors of flavored simple syrups can be pretty subtle. For a bolder blackberry taste, muddle fresh blackberries in the cocktail along with this syrup.
This syrup is even delicious when added to simple club soda or mocktails as well as cocktails.
Frozen blackberries can work in a pinch but I've found that the flavor is slightly reduced.
Many recipes add a tsp of fresh lemon juice to add a little brightness. I find that it can sometimes overpower the berry flavor. But feel free to try it both ways and let me know which way you prefer!
The lemon juice might also help the simple syrup from becoming crystallized. If you notice that the sugars are reacting with my recipe, make a note to add the lemon juice next time. I don't usually have trouble with crystals forming in my climate.
Some berries can stain! Take care when straining and transferring to a jar!