My dad uses facebook all the time on my PC.
The password keyring will remember only my accounts and not his (facebook account), it won't even ask to remember his password.
I want the Ubuntu 13.04 password keyring to remember his passwords instead of just mine. I don't understand why it remembers mine, but not his.
A keyring is never used to save a password (From Facebook or otherwise).
The Gnome or KDE keyrings are used to safely keep decrypted keys (hence the name) and no passwords.
One solution for you is to make use of two browsers. For example you could install Firefox and Seamonkey (because Seamonkey is very similar to Firefox.) You can also install Opera, Chrome, Konqueror...
The other is to check your existing browser and look to see whether someone clicked on "do not remember password" for that domain. If that's the case, then the browser stops asking you if you want to save the password. Once you have 2 or more passwords and you go to the login screen, it will no fill out anything until you start typing your email address (or other login credential, some websites ask for a username which increases security!)
With Firefox you can manage your passwords under:
Edit » Preferences ...
Security » Remember password for sites » Exceptions ...
You should see an entry for Facebook in that popup. If so, select it and remove it and then have your dad try again.
Note that by clearing all your history and other data, Firefox resets that scheme and it should ask you for your passwords again and give you the option to save them. Obviously, that means that in between you'll have to re-enter all your passwords...
