Someone calling themselves Hutton invented a new pen-and-paper cipher that seems to be incredibly secure. He offered a reward of over $1,000 to anyone who could crack it, although he disappeared from the internet a month ago so I'm not sure if the challenge is still ongoing.
Anyway, here is how it works. You come up with a scrambled alphabet and a key. Write your key repeatedly under your plaintext just like Vigenere. To encrypt the first plaintext letter, find it in the scrambled alphabet. Count to the right (wrapping if you reach the end) as many letters as the number which your key-letter represents (treating it as a base26 number where A = 0, Z = 25). The letter you land on is your first ciphertext letter.
Before you move on to encrypting the next letter though, scratch out those two letters (the plaintext and ciphertext ones) from the scrambled alphabet, and write them under each other. This effectively swaps them. For example, if you first plaintext letter is B and your first ciphertext letter is X, scratch out B and write X under it. Scratch out X and write B under it.
Here is a Javascript version of the cipher. Note that it treats password letters as numbers where A = 1 and Z = 26. This is only because I was creating it precisely according to the author's instructions and only realized the problem afterwards and couldn't be bothered to update it.
It would be very interesting to know if anyone can find any real weakness with this marvelously practical pen-and-paper cipher.