First make the ginger simple syrup: Pound the ginger into a paste in a mortar and pestle and add to a small saucepan with the sugar and ½ cup of water. Place on a medium heat and swirl the pan until the sugar dissolves completely. Remove from heat and let it sit for 10 minutes. Strain and refrigerate the syrup. Can be made up to one week in advance.
Place lychees, basil leaves, mint leaves, lemon juice and half the ginger syrup into a sturdy glass. Use a muddler, pestle or rolling pin to crush the lychees and fresh herbs. Fill the glass almost to the top with ice cubes and pour over the vodka. Top with a little soda water and stir well. Garnish with a few basil leaves and lemon slices and serve immediately.