Cut the cucumber slices down into smaller chunks. Throw all of the cucumber chunks and a small handful of mint into a cocktail shaker. Muddle them like you mean it for about thirty seconds, until there's significant cucumber juice and the mint leaves are very smashed and torn.
Add honey simple syrup, lime juice, and rum to the cocktail shaker. Fill it with a handful of ice, cap it, and shake vigorously for at least fifteen seconds. Break that ice up and crush that mint and cucumber even more.
Fill an old fashioned glass with a handful of crushed ice. Strain the drink over the ice, and top it with a bit of soda water for effervescence.
Garnish with a fresh cucumber slice and fresh mint leaves. Enjoy!