Here it is! I finally fixed the issue! I've literally tried everything and this is what i did! I downloaded the Gmail App and setup my email account. After that, i went back to a regular pre-installed Iphone mail app and added my Gmail account information and it worked. Basically, the app Gmail app from the Apple store somehow pushed the signal to Gmail, saying Here it is, it is working account.
I've spent about 3 days to figure out what was the problem, tried everything on the net, 2-step verification, Google display capture, deleting account and setting it up again, changing the password on Gmail account. I was going nuts! And i know there are so many people out there who are also having the same issue. They get this crazy message "Cannot Get Mail - The user name or password for 'Gmail' is incorrect." It was so annoying. I hated it!
The solution to this problem is: Go to App Store on your Ipad or Iphone and search for "Gmail". Download the first one, Free version, install it and setup an email account with your existing credentials, the ones that didn't work earlier, and to your surprise, it will allow you to add your account back.
Next step is to click Settings (on your Iphone, Ipad) --> Mail, Contacts, Calendars --> Add Account --> Gmail (Enter your lovely email credentials and you will see, it is working again, just like in early days!
I hope it will solve your frustration and disappointment.