Tools->Options, Security tab Passwords tab

Set a Master Password.

Then, every time you get asked for a password, enter it and mark the check that says "Remember Password"

You'll have to do it twice for each account (one for receiving e-mail and another when sending)

Close, open again, you should be asked for the Master Password once and then it should not ask you for any other passwords.

Not sure why it is being suggested you create a master password. That has nothing to do with saving password in Thunderbird. It is for restricting access to password you HAVE saved to Thunderbird.

In the password dialog box that appears and asks you for a password there is a check box in the corner that says use password manager to remember password. Check that box.

Well, don't use a master password and be the next guy who complains that his account has been compromised.

Good point but they are already complaining about having to enter a password every time they do something. This just adds another.

Well we aren't here to contradict each other, just make it easier.

Then, every time you get asked for a password, enter it and mark the check that says "Remember Password"

I've included that in the answer too.

No contradiction. Just clarifying your misleading post. If you stated that the Master password was a good idea but optional then it would have been more helpful.

In Thunderbird,

Tools|Options|Security|Passwords>Saved Passwords

Check that you have only one password saved for each server. If there are two passwords saved for an account, Thunderbird doesn't know which to use so prompts you to tell it the correct password.

A quick and dirty approach is just to delete all the stored passwords, forcing it to ask. There should be a checkbox on the dialogue where you enter the password, telling Thunderbird to remember the password.

If this doesn't work, there are a couple of files you could delete - but I don't know them offhand.