Make the syrup: In small saucepan, combine sugar, lime juice and grated ginger. Bring to a boil (be careful to not let it boil over), reduce heat and simmer a couple minutes. Remove from heat. Let cool. Strain and keep in refrigerator.
Make cucumber juice: Peel two cucumbers and dice in large chunks. Put chunks in blender and blend until smooth. Strain puree over fine mesh strainer into bowl. Toss solids and reserve juice in jar in refrigerator.
Make the cocktail: Add rum, ginger lime syrup and cucumber juice to a cocktail shaker with some ice. Shake well and pour in glass filled with crushed ice. Top with some club soda (use as much or as little club soda as you like. I used about 2 ounces).
Store left over ginger lime syrup in refrigerator. Use in more cocktails or try it in iced tea! You could also cut the amount of ingredients in half to make less syrup if you want.