Put blueberries & lime juice into a food processor and pulse until combined. This does not to be smooth since the signature part of a granita is it's grainy texture. If you are looking for a more refined finished product, then strain through a fine mesh sieve to remove the blueberry skins.
Next, add the blueberry lime mixture to a 9x12 baking pan, I used a glass one. Stir in the ginger beer and transfer to the freezer.
After 30 minutes take a fork and run it through the mixture to loosen it up, we are trying to achieve a grainy end product here and prevent a solid chunk of ice. Repeat at least 2 more times until you have a frozen kind of slurry result. If you don't use it all, you can transfer to a covered container and keep in the freezer, keeping in mind that it may need some additional thawing time before serving.