Place the sugar, basil sprigs and 1 cup of water into a small saucepan over medium-high heat. Cook, stirring every so often, until the sugar has dissolved, about 5 minutes. Remove from the heat and let steep for 30 minutes.
Discard the basil and pour the syrup into a pitcher. Stir in the lime juice and 4 cups of cold water. Add the blueberries and chill for at least 4 hours or until ready to serve.
To serve, put some ice in a glass, pour over the limeade and garnish with basil leaves. To make into a cocktail, add ice to a rocks glass and top with 1 1/2 ounces gin. Top off with the limeade and garnish with basil.