Also check that you are signed into Chrome with the account that you intend on being signed in with and that the sync settings are correct.It could be some weird sync issue where it is saving them until you close and reopen the browser or somesuch.
The key point of the Chrome OS is that you can store content on the cloud and access it everywhere.
When asked to save the password in IE or Firefox, say 'Yes'.
Now go back to Chrome and go to 'settings', now import saved passwords from IE or Firefox" worked for me.
For those of you who don't want to watch it, let me summarize it...
Chrome OS is the Google Chrome browser, nothing else special about it.
So I put in the full URL of "https://bowser.gruff.org/admin", and this time Chrome offered to save my password. Unfortunately I've got the same problem even though I checked remember my password and etc. I found a helpful link in the Chrome forum that suggests that this happens because your Chrome profile can become corrupted.