Password not remembered - Thunderbird
From MozillaZine Knowledge Base
Server is down
If a server is down, Thunderbird will forget your password and will keep asking you to re-enter it till it connects to the server or until you Cancel, whichever comes first. Sometimes when a server comes back up after a connection failure, Thunderbird will never log in to the server no matter what you enter. If this occurs, delete the perfectly good saved password using "Tools -> Options -> Passwords -> Saved Passwords -> Show Passwords" and restart Thunderbird (because the deleted password is still in memory). The next time Thunderbird prompts you for the password, enter it, and check the checkbox to save the password using the password manager. 
Several people have reported working around this problem with the Yahoo/Verizon SMTP server by replacing the mail server name with its IP address. Note that you will get an unrelated error (such as timeout or server unreachable) if you do this, and someday they will change the server address.
If one of your subscribed groups is write-only, as is normal for university assignment folders (for on-line students), the error that the server generates will prevent Thunderbird from EVER remembering your username or your password. One work-around is to unsubscribe to that folder, and use web-access to post to that folder. This is not too bad because it is usually only once a week or so that you need to post.
No checkbox to remember passwords
Instead of editing prefs.js, you can change the same setting via about:config.
Doesn't prompt for a password
The popstate.dat file is used to keep track of what messages have been downloaded from a POP server. Usually if it gets corrupted Thunderbird will download some of the messages again. However, several users have reported that Thunderbird stopped prompting for a password when checking for new mail and that they solved the problem by deleting the accounts popstate.dat file. You may want to move any old messages in the mail server's inbox to another folder using webmail before trying that since if you delete that file Thunderbird will download ALL of the messages in the inbox folder.
Two factor authentication
Check whether two factor authentication is enabled by logging into your webmail account using a browser and looking at the account settings. If its enabled you are probably supposed to be using an application specific password instead of the original password for the email account.  Two factor authentication is not recommended as most users (in our forums) who try it decide its not worth the hassle.
Check if the Signons.sqlite and key3.db files don't exist in the profile or have a recent creation date. CCleaner 3.04.1389 (64-bit) (a popular tool for automatically deleting unused files and orphaned registry entries)  has a bug where it actually deletes the stored passwords for all Mozilla applications when you tell it to delete Firefox passwords.
Usually whether or not you have a router makes no difference. But its possible you might need to clear its cache.