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Okay, so I looked this one up. I was curious about it, anyway.
It seems that when sailors or persons who were on a ship would become seasick, they went below deck, where the rocking motion of the ship was not so great and the stomach could settle.
The illness would most often happen when the weather made the seas rough, thus causing the deck to be more difficult to stand on.
Hence, one would be "under the weather" below deck, when they had to leave the deck and go below.
I always pictured a person who was said to be under the weather, walking around with a rain cloud over their head. Ha.