I dug out my old music theory notes, and I think this is what a Picardy third is:

"Picardy third - The term for the raising of the third, making a major triad, in the final chord of a composition which is in a minor key. The practice originated in c. 1500 and extended through the Baroque period."

I don't think Hotel California uses it, but I could be wrong. And I can't find out where it originated from either.