For the syrup: Combine 1 cup sugar with 1 cup water over low heat. Add a 1 inch piece of ginger, peeled and thinly sliced. Allow to simmer approximately 15 minutes. Remove ginger and place syrup in a clean jar. Will keep refrigerated for at least a week.
For the martini: Combine vodka, syrup and lemon juice in a shaker and shake until very cold. Strain and pour into a chilled martini glass rimmed with sugar. Garnish with a slice of lemon