I found this answer at this site:  Optimizing your PC   So, I haven't tried it or  had to use it, but the process seems plausible.

In some cases Outlook doesn't save your password, even if you enable the option to save the password. This error is caused by a malfunctioning of the Protected Storage Space from Windows. To solve this problem, run the registry editor and navigate to the key HKEY_CURRENT_USER\Software\Microsoft\Protected Storage System Provider. You will find a key with a long name starting with S-1-5-21 which you have to delete (probably it's save to create a back-up of this key by File, Export). In most cases the problem is solved and next time the passwords will be saved. Be carefull: by doing this you will lose all stored passwords. Read the back-up page how to recover these passwords.

If this doesn't solve the password issue (or if it's not possible to delete the key), there is probably a problem with the permissions of your user account. Select the key Protected Storage System Provider, right click on it and choose  Permissions, button Advanced. Make sure that your user account has Full Control for this key and subkeys. Activate the option Reset permissions on all child objects and enable propagation of inheritable permissions, click OK and repeat the earlier mentioned procedure. If needed, check the Microsoft KB files (http://support.microsoft.com/kb/290684). Here you will find more information how to reset the protected storage system provider.