<Which makes me wonder why a cucumber got picked for this distinction. Why not a watermelon? A watermelon is a lot cooler than a cucumber on a hot day.>

My gut reaction, which some gardening /literary historian (or more dedicated researcher!) will probably prove to be wrong, is that this is a traditional British English phrase which predates watermelons in the English-speaking world. I know you can grow cucumbers across a lot of the UK without a cold-frame, whereas I have no knowledge of UK-grown watermelons.

Alternatively, a quick google pulled up a couple of links:

http://www.vocaboly.com/forums/ftopic1884.html
and
http://www3.telus.net/jennybr/origins.html

which appear relevant even if somewhat speculative. And the image which springs to mind of an elegant lady having a facial with slabs of watermelon over her eyes rather than cucumber slices is rather appealing!