Although the notes indicate "...closed out the site and then attempted...", the key is to make sure the person is NOT signed in to the Event. (If the button at the top of the left nav says "Sign Out?", you're signed in.)

It's not clear from the note, but if this person has already fully registered and is trying to add family members through the family registration path, she has to begin with a new registration, i.e., Dad has to be a new primary registrant and then can add the kids. Mom's already registered, so she can't add secondary registrations.

/Rick