Place yogurt in an ultra-fine sieve or a strainer lined with cheesecloth and strain over a bowl in the refrigerator for at least 4 hours (or up to overnight). Combine yogurt, honey and vanilla in a food processor; pulse until smooth. Pour mixture into a squeeze bottle or liquid measuring cup; wipe processor clean.
Combine blueberries and honey in a medium saucepan over medium heat. Bring to a boil; reduce heat and simmer 10 minutes, stirring occasionally. Remove from heat and let cool for 10 minutes. Place blueberry mixture in food processor and puree until smooth. Pour mixture into a squeeze bottle or liquid measuring cup.
Fill popsicle molds with alternating layers of yogurt and blueberry mixtures, leaving a 1/2-inch of space at the top of each mold. Swirl layers together with a metal skewer. Insert popsicle sticks and freeze until completely firm, about 6 hours. Let stand at room temperature 5 minutes before unmolding.