Place the ginger, cucumber and mint in a food processor and finely chop. Transfer the mixture to a saucepan. Add the turbinado sugar, honey, water, lime juice and black peppercorns; simmer over moderate heat for 10 minutes. Strain into a bowl, pressing on the solids.
Wipe out the pan. Pour in the mixture and simmer over low heat until thick enough to coat a spoon, 10 minutes. Let cool; refrigerate in an airtight container for up to 2 weeks.