This randomly happened after we set up a slew of iPads with the same email account (Gmail on Mail), and sending messages put the outgoing message into the Outbox with an error that the password is missing, with the options to Cancel or go to Settings. Going to Settings, though, offers no option to input a password.

You can see here other users frustrated with this problem:
GMAIL Accounts unusable after 8.3 update!

Some workarounds offered are to use the Gmail app or to add Gmail as an Exchange account (if you’re using Google Apps and not a regular home Gmail account).

The workaround we employed was to double-click the home button, swipe away the Mail application, and then launch it up again. After a minute or so, there would be an error about the missing password again. This time, though, the prompt for Settings would take you to a screen to enter the password.

Very odd issue.