If your berries are large, especially blackberries, cut them in half. Place berries in a bowl and squeeze 1 tablespoon lemon juice over it. (Do not discard the peel.) Add honey, stir to combine. Set aside.
Place lemon peel, water and sugar in a saucepan. Cook, stirring, until it comes to a boil and sugar has dissolved. Gently simmer for 5 minutes more. Strain out peel and chill syrup completely (in fridge).
Whisk yogurt and chilled syrup together.
Place the berry mixture and any juices that have accumulated in a blender and whirl until desired texture.
Assemble popsicles by alternating pouring a little of the yogurt mixture, then a little of the berry mixture into each popsicle mold, repeating as desired until you reach 1/4-inch from the tops, leaving room for expansion as they freeze.
Use a fork to lightly swirl the mixtures together.
If using conventional molds, snap on the lid and freeze until solid, 3 to 4 hours. If using glasses or other unconventional molds, freeze until the pops are beginning to set (45 minutes to 1 hour), then insert the sticks and freeze until solid, 3 to 4 hours.