Wash cranberries and combine them in a pot with the water. Bring to a boil and let gently boil until all cranberries have popped and softened, about 15 minutes. Remove from heat and let cool.
Strain the cranberry juice into a clean pot, discard cranberries. Add sugar to the juice and bring to a boil again, stirring to dissolve the sugar. Remove from heat, let cool, then refrigerate until cold.
To make the ice, divide lime slices and mint leaves between the wells of a muffin tin. Fill each tin with water and freeze until solid.
To serve, pop the ice cubes out of the tin by running the bottoms under warm water until they loosen. Place in a pitcher and pour agua fresca over the top.