You have to enable this feature in Privacy settings. You can get there by going into Chrome settings:

Enable saving passwords

1. Click the Chrome menu icon (the three dots on the top right).


2. Click Settings
   


3. Click Show advance settings (new the bottom of the page)


4. (Under Privacy - Passwords and forms)Place check marks under the following:
   
   • Enable Autofill to fill out web forms
     
   
   
   • Offer Save your web passwords
   
   

Passwords can be saved even with cookies disabled. You can still check your cookie settings with the steps below.

Enable or disable cookies

1. Click the Chrome menu icon


2. Click Settings
   


3. Click Show advance settings
   


4. In the Privacy section, select Content settings
   


5. Select Allow local data to be set
   

If all settings are set for saving passwords and it still fails check the integrity of Chrome's setting by bring it to the defaults with the following steps.

First exit out of Chrome. Then run these commands to temporarily remove the current settings.

$ cd ~/.config
$ mv google-chrome google-chrome.bak

You can get the previous settings back by renaming google-chrome.bak to the it's original.

The issue was resolved as detailed in the chat comment section by renaming the home directory to a backup name, then creating a new home directory where Google Chrome worked. Then the backup directory was migrated to the new space.

Workaround

If above answer does not solve the issue, installing the beta version of Google Chrome can workaround the problem.

apt install google-chrome-beta 

This seems not to solve it (entry preserved for historical reasons):

I have the very same problem and did a few hours of debugging.

My problem is only related to google-chrome-stable Version 53.0.2785.101 (64-bit) on ubuntu 16.04.

When I login at a website, cookies are set correctly. When I close chrome, log out, log back in again, the cookies are still there and everything is fine. However, if I reboot, the cookies are gone and I need to login again.

I did some debugging (reinstalling chrome, reverting to default config, starting from a clean config, everything as suggested above, ...) and the bug persits.

I installed google-chrome-unstable Version 55.0.2853.0 dev (64-bit) and everything works fine (while stable still has the issue). So this is probably a chrome software bug.

My "solution": Use firefox/chrome-unstable for a few days/weeks, just wait for the update which fixes the issue to propagate to google-chrome-stable.

By the way, I'm using ubuntu 16.04 (systemd) with dm mapper/cryptsetup which was upgraded from 14.04 and I'm still not 100% sure that my crypt partition is unmounted every time 100% correctly (though the fs is always clean). @Novah, do you have a similar setup?

It is September 2016 when I write this advice. If it hasn't resolved within a month automatically, my answer is wrong ;-)
I hope to spare you a lot of debugging and frustration.

Please note this is already reported to the Chromium Bug Tracker: https://bugs.chromium.org/p/chromium/issues/detail?id=631171

I had the same problem, but my system is ubuntu 14.04.

If your system also asked for keyring password every start google chrome (also restart system), i think that because Google chrome cannot get your saved google-password because your saved google-password locked by the keyring. You have to make sure that keyring is unlocked before you start Google chrome, not after. Here is my solution:

1. Open google chrome.


2. Login back to google account as usual and make sure there is no yellow warning icon before your username at top right corner. (just do it)


3. Close google chrome.


4. Open up Dash and open User Account, then turn "off" automatic login.


5. Restart system.

By turning automatic login to "off", I thought this automatically will unlock keyring every login system, so Google chrome can get your google account password from keyring.

I know there are also another solution about disable keyring (by set keyring password to blank), but i think that is not secure.

I was facing same issues. In new install, I installed chrome as root ( not sudo). After restarting the program worked well.
Learning this, I uninstalled chrome from the previous PC and installed it again as root user.
Now everything is fine and the annoying login prompt is gone.

sudo -i
apt-get remove google-chrome*
apt-get autoremove
apt-get clean 
apt-get install google-chrome-stable 

I have encountered the same problem. For the issue, it is because of the automatic ubuntu login.

So, for the easiest fix, just disable the automatic login.

Disable the automatic login for Ubuntu:

Ubuntu disable automatic login

Disable the automatic login for Ubuntu:

• Open terminal




• Enter, sudo nano /etc/lightdm/lightdm.conf




• Comment your username,

#autologin-user=MyNameIsSyirasky

• Save and exit, reboot.