Hi
“Well,” said the inspector, refastening his notebook. "The case seems clear enough.
© Agatha Christie
So, notebook here is just a notebook. 
(To me, it always takes some time to switch to that time, despite the fact that in the UK laptops are more commonly used :D)
But refastening in this context?
Anyway, to me it sounds very strange to say fasten a notebook.
What on earth did he do with his notebook? :lol:
Some notebooks or diaries have a kind of closing/fastening device, like a press stud, for example. Some can be ribbon-tied or even locked.
Probably nothing more complicated than that.
Thanks, Conchita.
There is no detail about the inspector’s notebook (what I gave you is the first its mentioning in the novel), but perhaps, you’re right. Nothing else to suppose…
Maybe that time Scotland Yard’s inspectors had some special notebooks and readers were/are well aware of that. But not me. 