Persistent "Google sign-in" error, even with correct password
Here is the solution: Login into Gmail, locate Account under the sign-out Name. This takes you to My Account where you click on security. Here you find a section for Application-Specific passwords at the end of the webpage. The issue is explained that:  Some applications that work outside a browser aren't yet compatible with 2-step verification and cannot ask for verification codes, for example:

Email programs including Outlook, Apple Mail, or Thunderbird
Older Android smartphones


The root-cause of this problem is that you probably turned on the 2-steps verification without noticing this. So all you have to do is to generate your application specific password and that is it!