If you look to the right of your post you will see "More like this" list of similar problems. To wit the error message indicates that you are trying to copy / move files during import to a folder location where you do not have full read and write permissions.
I've been reading in the forums and many people are claiming this is due to a file permissions issue. The is not the case here. I am logged in as myself, running Lightroom as myself (the Lightroom process is owned by me) and my entire home directory is owned by me, including the "2015/10" folder it would want to write into. In addition, I can manually go into this directory and manually create and delete files and folders without any problem.
Doesn't matter - Lightroom is being prevented from writing to the destination, and only the OS can do that.
The error was due to file permissions after all. When importing media, you have to look very carefully at the import gear menu defaults - if you have "Into Subfolder" selected, look carefully at the setting for "Destination Folder" - in my case, it was set to "/". This is the root directory so it's obvious standard user level permissions will not allow files to be written there, nor did I want to have anything saved there. It was especially easy to miss since it was only a single slash. Once I set the destination folder to a location within my home directory, the import worked fine.