Take a deep breath, let it out.
When you scan the documents, you scan them as a group (so you could actually scan multiple groups into a note). When you select "keep scan" after snapping the scan and possibly editing it, You see a "save" button. Click the button to add the scans to your Note (you could also take another scan here). The scanned documents then appear in your Note. with a title, like "scanned Documents". click the title to view, and click the title in the view, ergo "scanned documents", you will see a "rename" dialog. Trust you can handle the rest.
BTW: tap the share button to send, save Create PDF, print, etc........ , and click the image to edit documents, you can also click the "+" to add more scanned documents.
oh and you don't rename it by changing the note title. This wouldn't work if you had more than one group of scanned documents.
Nothing could be simpler. If you think about it, if it automatically came up with a Rename or save dialog, that would be an extra step when you don't need to do that, and no extra steps when you don't
That's not to say your comment is not relevant, there could be a Rename option on the share menu, but with the name of the file right next to the share button, there kind of is.