In a large non-reactive saucepan, crush blueberries with a potato masher or back of a heavy spoon. Add vinegar, sugar, lime and cinnamon and bring to a boil. Reduce heat and simmer, covered, for 20 minutes. Cool slightly and pour into a large bowl. Cover and refrigerate for 2 days to allow flavors to blend.
Place a wire mesh strainer over a large bowl. In batches, ladle blueberry mixture into strainer, pressing out as much liquid as possible. Discard solids.
Pour vinegar into clean glass bottles or jars, and refrigerate, tightly covered, indefinitely. Use in salad dressings or drizzled over chicken or beef.