1. Thinly slice 1/4 of the cucumber and set aside. Cut the remaining cucumber and put into the bowl of a food processor and whirl with 1/4 cup water until well blended, about 2 minutes. Pour into a strainer set over a bowl. Let sit 10 minutes to drain, occasionally pressing with the back of a spoon to push liquid through; discard solids.
2. Put cucumber juice, reserved slices, mint, simple syrup, orange and lime juices, and orange blossom water into a container to chill, at least 30 minutes or up to 1 day ahead.
3. When ready to serve, add ice and top with club soda.
*This adds a nice citrusy fragrance. Find it at Middle Eastern or well-stocked beverage stores.
Total time includes chilling.