Originally Posted by coth
I also had the same problem. I believe the problem is related to issues in the User Account profile (User Account with administrator privilege and a password).
I actually solved the "Windows Mail Won't Remember Passwords" problem when I was fixing a 2nd problem, "Windows Live Messenger error 80070005". I tried deleting/adding the Mail Accounts to no avail. I also previously changed my User Account from Administrator to Standard and back to Administrator, to no avail.
I finally fixed the problem as follows:
(1) Removed the password from my User Account
(2) Restarted the computer
(3) Added the (same) password back to my User Account
(4) Restarted the computer & Logged into Windows Mail
(5) Updated my Mail Accounts using Tools > Accounts > Properties. I added the password and ticked Remember Password. Windows Mail remembered the password from this point on (problem solved).