Check out Manuel Blum's human computable hash function. He calls it HCMU for Human Computable Machine Unbreakable.
He claims you have to spend an hour memorizing the technique and then you can apply the has function in about 20 seconds without even using pencil and paper.
The memorization required is to remember a random mapping of each letter of the alphabet to a digit in the range 0 to 9. You also need to remember a random permutation of the numbers 0 through 9. These stay the same for each string you want to hash so you only have to memorize them once. It's well within the ability of the average person to remember.
The hash function involves mapping the input string to digits, doing simple addition functions and then looking up values in the random permutation you memorized. He has a proof that it should be difficult for a computer to break the hash.
http://www.heidelberg-laureate-forum.org/blog/video/lecture-monday-september-22-2014-manuel-blum/