Why do we use cucumber to describe the attitude of a person who is able to control his/her temper in front of a difficult event?
This entry in the Encyclopedia of Word and Phrase Origins (p133) has the answer:
It took scientists with thermometers until 1970 to find out what has been folk knowledge for centuries - that cucumbers are indeed cool, so much so that the inside of a field cucumber on a warm day registers about 20 degrees cooler than the outside air. The belief is ancient, but was first put on record by Francis Beaumont and John Fletcher in their play Cupid's Revenge (1610), when they referred to certain women as "as cold as cucumber".
In addition to Shoe's answer, cucumber slices over the eyes are a long-standing folk remedy, rejuvenated into a modern luxury spa treatment. If you've never tried it, they really do have a soothing, cooling effect even if they weren't previously refrigerated.