Use the remaining honeydew for smoothies, or blend with cucumber, mint, onion, and a jalapeño pepper for green gazpacho.
Using a vegetable peeler, shave 1 quarter honeydew into long strips to yield 1 cup (reserve remaining honeydew for another use). Shave cucumber into large strips to yield about 1 cup.
Combine shaved honeydew, shaved cucumber, and remaining ingredients in a bowl; toss gently to combine. Serve immediately.