In a medium saucepan stir together water, sugar, blackberries and ginger. Bring mixture to a boil, then reduce heat and let mixture simmer until sugar completely dissolves. Remove from heat, cover, and let steep for 15 minutes.
Strain blackberries and ginger from syrup mixture and let cool completely.
Assemble the bellinis by adding the sparkling wine to a champagne flute, filing halfway, top off with the blackberry-ginger simple syrup. Garnish with fresh blackberries and thyme sprigs, if desired.