Make a simple syrup by boiling sugar, water and ginger. Remove and chill. Peel and chop cucumber and puree in blender with strained, chilled syrup until smooth. Strain puree through a chinois and place in a metal pan and put in freezer. With a fork, scrape granita around all edges and mix well every 30 minutes until it reaches icy consistency. Season to taste with lime juice and salt and pepper.