Starting to get extremely annoyed at this now.
I have no idea what to do or why this is happening.

Issue: Unable to login to Dashboard/Wordpress admin the second time.

When going to /wp-admin the first time after install I get graphical login:
Like this one: http://www.webtutorialplus.com/wp-content/uploads/2013/05/wordpress-admin-login.png

I can then edit site, change themes, etc. Do everything I need to do.

Next time I visit site, I get a popup instead which just states "http://mysite.com:80" "Restricted area" and a form for User: and Password:
Whatever I type here it says 'unauthorised' and I can't get in.

Solutions tried:
-Reinstalled Wordpress, works once, then not again.
-Reinstalled Droplet, works once, then not again.
-Changed user using WP-CLI, do not help
-Changed password using WP-CLI, do not help
-Reset everything in Wordpress folder, can login once, then never again.
-Tried from several different computers.
-Tried different networks to rule out firewall issue.
-Tried different browsers.

Setup:
-I use Ubuntu 14 with Wordpress One-Click-Install.
-I have my own domain. I use a subdomain fro the blog.
-The host name of the droplet is subdomain.domain.com
-My DNS is pointing A-record from subdomain to IP of droplet.
-I use subdomain.domain.com/wp-admin to try to login. But same issue with /login and /admin