Symptoms

When runnng an Office 365 backup of the SharePoint Site collection you get the error message:

SharePoint Online: failed to get the lists from (https://domain.name/sites/user) : The remote server returned an error: (401) Unauthorized..

Cause

The user account that you are using has not been configured as a Site Collection Administrator for each of the sites that need to be protected.

Solution

In the SharePoint admin center, expand "Admin centers", Click on SharePoint, the list of available site collections is displayed, then place a check mark on the intended site(s) and click Owners tab, followed by Manage Administrators to add the user accounts that will be used to do the SharePoint backup sets to the Site Collection Administrators group. 
Site Collection Administrators are given full control over all Web sites in the site collection. They may also receive site use confirmation mail. Enter users separated by semicolons.