Slice cucumbers down the center lengthwise. Sandwich mint between cucumber halves as you feed it through a juicer.
Remove rind and seeds from honeydew and put through juicer.
Combine cucumber, mint, and honeydew juices with lime juice and sugar. Adjust to taste with more sugar or lime juice as desired.
If juice is to be stored for longer than a day, add Ball Food Protector (found in the canning section of the grocery store) to preserve color and freshness.