OK. I found out in the end.
Authors can upload files into the media library, and that’s where the weakness was. He uploaded a particular file that roots around the website and find weak points and essentially hacks it. From this, he could basically set up the website as if it were being installed for the first time, and create an admin account as the creator of the website.
He’s a little cheeky monkey, but I did get to the source of the issue. Basically, he was only able to do this because he already had Author access to the site. He couldn’t have uploaded a file to the media library if he had just been a subscriber.
Since then though, I’ve installed a raft of security plugins, only let .png and .jpg to be uploaded, and as a rather cool Dual-Authentication SMS code generator, which basically texts me a code every time an admin tries to log in (very cool).
All of which means that he won’t be able to sneak in again.