For what it's worth, it definitely is "Here here!", and not "Hear hear."
But for some reason most people get it wrong and the other way around. Not sure why.

I am unconvinced this is right. I always assumed it to mean a shortening of hear ye year ye..and a quick google search shows several sites saying hear hear is the correct spelling..I don't have time to do an exhaustive search. Perhaps someone can but I do not believe here here is right. mg