First we are going to make a cucumber simple syrup. Combine the 3/4 cup sugar and 3/4 cup water in a small sauce pan along with the one cucumber. Warm it over a medium-high heat and simmer until the sugar is dissolved--this makes 1 cup total simple syrup.
Once the simple syrup has cooled to room temperature, strain out the cucumber and transfer it to the refrigerator to chill. NOTE: You might want to make more than just one cup. This syrup would be great in many beverages, like a limeade. The general rule for simple syrup is that you need 75% of both sugar and water for whatever measurement you want. Since I wanted 1 cup total for this recipe, I used 3/4 cup of each.
Make your green tea but make sure to chill it before using it in this cocktail. You don't want to serve your guests a warm mojito, right? I am assuming that you are going to share this four-serving recipe with others. Am I wrong?
Add five of the mint leaves to a cocktail mixer and the juice from half of a lime, then muddle the mint. Now, don't beat the mint up. Muddling is about releasing the essential oils of the herb into the drink. You don't want to go all Ike Turner on it.
Now add the remaining juice from that lime, a heaping 1/3 cup of rum, 1/4 cup of green tea and 1/4 cup of the cucumber simple syrup. This is going to be a very smooth mojito, by the way, and very refreshing. Play with the green tea and cucumber measurements to take it in different directions. Or just exclude one? You cannot go wrong.
Mix and serve over ice with a splash of club soda, cucumber slices, more mint and a lime wedge.