In a small saucepan over medium-high heat, combine the blueberries, thyme stems, sugar and water. Cook until the berries are bursting and become very soft, around 5 minutes. Place a mesh sieve over a small bowl and push the berry mixture through to remove large chunks of skins, seeds, and stems. Cover and refrigerate until ready to use.
To make the cocktail, fill a glass with ice. Add the blueberry thyme syrup, followed by the Seagram's Extra Dry Gin, and top with the lemon-lime soda. Stir gently to combine, then garnish with fresh blueberries, thyme sprigs, and a slice of fresh lime. Enjoy responsibly.