Add the candied ginger and the water to a small saucepan. Bring to a boil over high heat. Turn the heat off, and let the ginger sit in the hot water for 30 minutes.
Transfer the candied ginger-water mixture to a blender. Blend on high speed for 30-60 seconds until smooth.
Pour the ginger mixture through a fine-mesh strainer into a bowl or large (4-cup) measuring cup. Use the back of a spoon to press all of the liquid out of the ginger solids left in the strainer. Discard any remaining ginger pulp.
Stir the lime juice and sugar into the strained ginger liquid. Transfer the mixture to a large airtight jar or bottle and refrigerate until fully chilled.
To serve:
Pour over ice and add sparkling water to taste to make a delicious non-alcoholic cocktail.Pour over ice and add 1-2 ounces of vodka and sparkling water to make a Moscow Mule. Serve in a copper mug.Pour over ice and add 1-2 ounces of dark rum and sparkling water to make a Dark and Stormy.
Notes
I typically find candied ginger in bulk food stores and departments. It may be available in the baking aisle. It's also widely available online. It may also be called crystallized ginger.