Combine the sugar, thyme, and 1 cup water in a small pot. Bring to a boil and cook, stirring occasionally, until the sugar is dissolved. Let cool and discard the thyme. Combine the thyme syrup with the strawberries, lemon juice, and 5 cups cold water in pitcher. Chill for 30 minutes or up to 12 hours. Add to beverage jar, top with ice and garnish with fresh thyme.