The main difference between what you want an example for (digital signatures) and secure communication is this: the roles of the public and private keys are reversed.
Also, the content being encrypted is different. For secure communication, the entire message is encrypted. For digital signatures, the message format is irrelevant; you are trying to prove authenticity, not protect message content. Rather, it is a digest, hash, or checksum of the message that is encrypted.
The signature algorithm computes the digest, encrypts it with the private key (known only to the sender), and includes this encrypted digest (the signature) along with the message, to prove its authenticity.
The verification algorithm takes the public key (known to everyone) and uses it to decrypt the signature, as well as computing the hash or digest of the message itself. It then compares its computed hash of the message to the decrypted signature. If they are identical, verification was successful. If not, verification fails.
This is in contrast to security usage, protecting the content of a message. In this case, the message is encrypted using the public key of the recipient, and thus can only be decrypted by that intended recipient who has the matching private key.
So if you have an existing pen and paper example for security purposes, you could just use that same example, reversing the roles of the private and public keys. It's just a matter of understanding the differences in the process.
The only additional piece is the digest or hash. You can choose any mathematical function for this; a checksum or simple equation would be fairly easy to compute by hand. A checksum for example could be used to make a digest this way:
We can make it simple: A = 1, B = 2, etc. So we have
H = 8
E = 5
L = 12
L = 12
O = 15
The checksum is then computed by multiplying the position of each letter by its value, and summing them all up. So now we have
checksum(HELLO) = (8*1)+(5*2)+(12*3)+(12*4)+(15*5)=8+10+36+48+75=177
Now we encrypt 177. If our encryption method is xor (to simplify things on paper) and our private key is 915, the xor result is
Simplify using place values:
Simplify again using binary:
=(0001^1001) | (111^001) | (111^101)
Now we can easily compute the binary xor operations and add the results:
=1000 | 110 | 010
Back to decimal:
So we send the message
HELLO862 probably along with some way to specify the message length. In most cases the signature might not be distinguishable from the message itself without knowing in advance how long the message is.
Then take 862, the signature, back through the decryption process. In our case we used xor which is a symmetric encryption, so it sort of defeats the purpose, but when using RSA, your private key and public key are two different numbers.