This drove me nuts for awhile. I have Google Apps so my GMail account is not .gmail.com.  No matter what I tried, I could not get Outlook to connect to my GMail account which has 2-step verification turned on. After doing quite a bit of research I finally figured it out.

You need to set up a separate password for accessing Google Apps including GMail.  So going through the normal procedure you would find on the web will not work, including the steps outlined on Outlook and Google’s own site.  To make this work, you need to go here..

https://support.google.com/accounts/answer/185833?hl=en

Follow the steps and you will get a new password.  Then start the account process over again within Outlook.  BE SURE to remove any selections that save your password.   Enter the new password and Outlook should start working with your GMail Apps account.   Once this is working correctly, then you can turn the remember password options back on again.

Hope this helps.