I was thinking they could add it to the end of the private key and but I don't think that is how RSA works.
Indeed it is not.
How could the decrypt a say RSA-encrypted message using this information?
One solution is to encrypt a private key with the password and encrypt the message under that private key. This also allows you to add new messages / remove old ones w/o having to recover the private key first. Of course if you'd always do that anyways, you could also just forgoe the asymmetric encryption and just use plain password-based encryption.
The cleaner solution would of course be to encrypt the message under everyone's public key, but given that it sounds like the family is not so much up to tech, a simpler solution like the above one will probably be better.
Of course note that the security of passwords like a dog name is very questionable and for a proper security level one would probably have to combine a dozen or so such information pieces (assuming that every question has ~2000 equiprobable answers).
[What] [...] would be the easiest [way] for someone with limited computer knowledge to decrypt a RSA encrypted message?
Using a standard tool and writing a detailed guide (ideally with pictures) on how to use it. A GUI is probably preferred as quite a few people are scared of command lines.