How did the expression "under the weather" come to mean feeling ill?

Answer 1

The expression is indeed of nautical origins, but it seems to be from the practice of going down below deck when one was feeling seasick (rather than being out in bad weather).

http://see-the-sea.org/nautical/naut-body.htm#U
http://www.dauntlessprivateers.com/nautical_sayings.htm
http://www.phrases.org.uk/bulletin_board/28/messages/325.html

Answer 2

It comes from standing quarter deck watch in a storm. You are out in the wind and rain and on an active section of the boat feeling the action of the waves.