Place sugar, water and sliced ginger in a small saucepan and bring to a boil, stirring to dissolve sugar.
Reduce heat to low, and simmer ginger for about 30 minutes, or until tender.
Strain ginger, reserving syrup.
Pour syrup into a drink container with a lid, and refrigerate until cool. Syrup can be made several days ahead.
For martini, mix all ingredients with a few pieces of ice, in a martini shaker, and pour into 2 glasses.
For mocktail, stir ingredients together and pour over glass with ice.