The Team Site: Notebook feature allows you to link a OneNote notebook that is shared amongst the team on your site.
Symptoms
When migrating a site that uses this feature, you will find it does not work at the destination.
Details
Though we do migrate the actual OneNote notebook, the unique ID of the Notebook changes in the new environment, and that breaks the link to this feature.
Solution
You can follow these steps to manually link the OneNote notebook at your destination site:
Click Site contents.
Go into your Site Assets.
The OneNote notebook you migrated will be in this folder.
Duplicate it's contents into a new OneNote notebook, and place it in this folder.
You will now have the old OneNote notebook that was migrated from the source site, and the new one you just created.
In the side bar it is possible to edit the links, we recommend switching to classic SharePoint since it's easier to modify the links.
In the classic experience click EDIT LINKS on the sidebar.
Drag and drop the new OneNote notebook to the sidebar.
Rename the new link.
It will now be possible to delete the old OneNote notebook link that is not working.
Once done, the feature should work properly.
Tip: You can refer to this TechNet forum for useful information on syncing OneNote files after your URLs change.