How can I check the size of a OneNote notebook on iPad?
Hi if you've taken the path of least resistance and defaulted to using Microsoft's OneDrive for storing OneNote documents, 

then you can follow this route:

First make sure that OneNote documents you are interested in are synced.

On the ipad's onenote screen the 2nd icon in on the purple ribbon is a notebook,
touch that and select 'sync this notebook'
This ensures a copy is with stored on the One Drive.

Now you need a separate Apps store app to look at the OneDrive, which handily is called OneDrive.
Download that from the app store,
login to your onedrive,

You'll now be able to see your OneNote document, and the size is shown by default.