TB doesn't have the option to change passwords. They can only be changed on the provider's website or by phone. Once the new password is confirmed working, delete the old ones from Saved Passwords in Preferences, restart TB, enter the new password when prompted.

Once you get to the privacy&security section and are viewing passwords, doubleclick the one to change. Password can also be set at tools>accountsettings>Outgoing Servers. That changes within TB, but only at server can password be reset.

I had to change my email password at Comcast. But for some reason the old password seems to have completely disappeared from Thunderbird. That is, it is not listed in the SAVED PASSWORDS list at all; in fact there are no passwords in that list. This means I can't doubleclick to change it and there doesn't seem to be any way to add a password. I would like to put the new password into settings so I don't have to enter it every time I want to get or send mail.

On the password page, you must first click the button to show passwords. Then, right-click on the appropriate row and a menu opens up to edit password.