We have a local server to which I was trying to connect - "https://bowser.gruff.org/admin".
When I first connected, I left off most of the URL, just entering "bowser/admin", which worked (because of sufficiently satisfactory name resolution). However, I got a warning from Chrome (Chromium, actually) that warned that the https site was not secure; I just went into the Advanced settings and told Chrome to ignore the security warning, and was able to log in. But Chrome never prompted me to save my password.
After a couple of days of this, I came and found this site, and after trying other things unsuccessfully, reset my Chrome settings to the default ("Settings" / "Show advanced settings..." / "Reset settings"). Then when I reloaded my page, I again got the security error, and noticed the URL address bar had a red flag with https X'd out, and realized maybe Chrome is not saving passwords on insecure https sites.
So I put in the full URL of "https://bowser.gruff.org/admin", and this time Chrome offered to save my password.
Hopefully this info might help someone.