Special equipment: a 2 1/2 cup-capacity martini shaker.
In a pitcher, combine the syrup, gin and lemon juice. Refrigerate for 2 hours. To serve, add ice and half of the gin mixture to a martini shaker. Add 2 tablespoons of blueberry jam. Shake well and pour into ice-filled Collins glasses. Add a splash of club soda and garnish with fresh thyme sprigs. Repeat with remaining ingredients.
Gently crush the thyme sprigs and put them in a small saucepan. Add the sugar and apple juice. Bring the mixture to a boil over medium heat. Reduce the heat and simmer, stirring occasionally, until the sugar has dissolved, about 5 minutes. Remove the pan from the heat and allow the syrup to cool, about 30 minutes. Strain before using.