Cannot update Gmail password in Mail

I was able to resolve my issue.

I found out where to enter my password and the password field was empty. Here is the location for where to put your password in Apple Mail:

Under Mail > Preferences > Account > "Your Gmail Account" > Account Information, click on Outgoing Mail Server (SMTP) and then select "Edit SMTP Server List...". Make sure your account is selected and then click on "Advanced". There should be a place to put in a password.

After putting in my password, I now got a "Login Failed" error and a pop-up window prompting me to take all account online. This didn't work. What I had to do was log back into google and enable the use of "Less secure apps". It seems that turning on (and then off) 2-step verification also turned off the use of less secure apps, which I had previously had turned on so I could use Mail.

Maxplanar, I came across this thread that might solve your problem: How to update GMAIL Password on MAIL.

Under Mail > Preferences > Account > "Your Gmail Account" > Account Information, click on Outgoing Mail Server (SMTP) and then select "Edit SMTP Server List...". Make sure your account is selected and then click on "Advanced". There should be a place to put in a password.

I am having a very similar problem. I am using the most recent version of OS X El Capitan (10.11.3) and Apple Mail Version 9.2 (3112). Today I decided to enable 2-step verification on my Google account. After enabling it, the lightning bolt symbol displayed next to the gmail mailbox in Apple Mail. This was expected since I had not yet set up an App Password to use with 2-step verification. I set up the App Password and was given a new password to enter into Apple Mail. Contrary to the instructions, there was no place to put a password in Apple Mail under Mail > Preferences > "My Gmail Account". I have since turned off 2-step verification but the lightning bolt symbol is still displaying next to the mailbox (and I am no longer getting mail).

NOTE: My gmail account in Mail was working fine before enabling and then turning off 2-step verification. I never changed my gmail password in Mail, and my gmail password still works when logging into gmail from Safari.

I have tried logging out of my Mac several times and restarting Mail, but the symbol is still not going away. Along with the lightning bolt symbol, in the top right corner of the Mail window there is an error that reads "Login Failed". I find it quite interesting that I get no pop-up windows prompting me about the account (could not connect/enter password/etc).

In Mail, I opened Window > Connection Doctor and there are red circles next to the gmail IMAP and SMTP account types indicating that these account could not connect. The following error is listed next to each account: "Trying to log in to this IMAP/SMTP account failed. Verify that the username and password are correct." I clicked "Check Again" and there was no change.

I also checked to see if I could reenter my password in System Preferences under Internet Accounts > "My Gmail Account" but there is no place for a password.

Desperate to solve this, I referenced this threadand then opened Keychain and deleted the imap and smtp passwords for the gmail account, thinking I would be prompted for my password when I tried to bring the account online in Mail. However, everything remains the same. I have searched Google and cannot seem to find a thread that discusses anything about this issue.