In response to @Aaron's responseanswer, there is Manuel Blum's Mental Hash Function, but I did want to say that Rob Shearer has shown that it is machine breakable if you know that someone is using the function and you have 150 characters of ciphertext:
The "Blum Mental Hash" Is A Lousy Idea
Having said that, I am practicing to use it by memorization, and I do think that it fits your criteria. I have been practicing to use the algorithm for about a week, and I am now able to hash 10 character words in about 10 to 20 minutes without using pen and paper.
Still, in my opinion, it is insecure as a hash, and I am still evaluating using it, but I also don't plan on making any of the ciphertext public, which I don't know if this is a form of "security through obscurity," or not.