Combine boiling water and tea bags; stir once, and steep 10 minutes. Pour through a fine wire-mesh strainer into a pitcher to equal 2 cups hibiscus tea. (Do not squeeze tea bags.)
Process blackberries in a food processor until smooth, about 30 seconds, stopping to scrape down sides if needed. Pour through a fine wire-mesh strainer, stirring and pressing with a spatula to extract as much puree as possible. Discard seeds and solids.
Stir together hibiscus tea, cold water, honey, and 1 cup of the berry puree in a pitcher; serve over ice. Reserve remaining berry puree for another use.