My experience with the change password system of Age of Conan sites was frustrating to say the least. However, unlike other poor souls who have had weeks or even months of being unable to log into the game after changing their password, after 2 days of researching plus trial and error the solution I have found is actually embarrassingly simple.
Step 1 - Go to this site: https://account.ageofconan.com/
Step 2 - Click “Forgot your password?”
Step 3 - Enter your User Name or Email
Step 4 - Go to your email and copy the randomly generated password that they assign your account.
Step 5 - Launch Age of Conan and enter that password upon reaching the Username and Password screen.
You can now play Age of Conan. Leave the randomly generated password alone and use that from now on. Just save that password in Notepad or something so that you can copy and paste it. If you try to change the password to something you prefer you will find yourself in the same situation and will have to repeat these steps.