Sys Internet Accounts keeps asking for gmail passwords

Do you have any other email accounts listed in Mail (besides gmail) [Mail > Accounts ]? Since you could have other email accounts, the server (sending/receiving) settings within Mail could have remnants of your gmail account there (possibly).

You don't need to delete all of the accounts. If it says it already exists it could be that your gmail smtp outgoing server setting was used in one of the other accounts. Select Mail > Preferences and see the list of accounts and go account by account to see that none of you non-gmail email accounts reference the gmail smtp as outgoing. Apple support also suggests to rebuild mail [Mailbox > Rebuild]. Another suggestion to ensure that Mail itself is not corrupted, by using another user login account on your computer and add the gmail account there to see if it works (you can always delete).

Since gmail accounts are IMAP you can try deleting the email account from Mail and re-create the account -- that did the trick for me. Steve Fortier above tried that and it didn't work when adding the account back. He was able to restore Mail from a backup (presumably Time machine?). The issue there is how old is your last backup and if it results in lost emails from non-IMAP email accounts (i.e. POP accounts).

Before I deleted my gmail accounts (see the entire thread above), I followed the other suggestions of deleting the gmail passwords in keychain etc. So do those steps first and see if that clears it up (then delete the accounts and add them back). I also did the combo update as suggested (although there has been a new software update since then, so you may need to contact support for the latest version if you want to try that option).

Where are you trying to delete the account (system preferences > internet accounts via Mail > Accounts or through Mail > Preferences)? Try Mail > Preferences and then select the gmail account listed in the account pane and then click on delete ( - ). Then go to Mail > Accounts to see if its still listed in the Internet Accounts list.

That's very strange. Try repairing permissions. Open Disk Utility from Utilities, highlight your hard drive (Macintosh HD) on the left and select "repair permissions" from the options/selections. It can take a while to finish. Then you can try deleting.