After restarting the website and redoing some of the steps it all works. It seems there was some caching issue that was stopping me from being able to add items.
I'll try to explain what I want to do (which now works, but only after restarting the website).
Created a book using the Chapters module named "MyBook".
Created two chapters under the book: "MyChapter1", "MyChapter2".
When I created the book, the "MyBook Chapter" Content Type was automatically created for me (by the Chapters module I assume).
I then edited the "MyBook Chapter" type and added the "Container" part.
I created a content type named "MyEntry" and added the "Body", "Common" & "Containable" parts to it.
What I expected after this was that I would be able to create a new "MyEntry" item and the "Add to" dropdown would display the chapter, so I could then add the new item to the chapter. This step didn't work. Nothing was displayed in the dropdown list except
for (None). However, after I restarted the site this was now populated with my chapters and I can successfully add the entries.