How to change the color of new highlighted message

Even though there is no option to edit chat window messages by default, you are able to customize them using Custom CSS. To do so, navigate to Configuration > Chat > Chat buttons > Edit your chat button. Then in the "Chat window" tab scroll down to the "Custom CSS" section and add the following CSS code:

.MessageNotification.MessageNotification-Unread {
    background: green !important;
    color: white !important;

Do not forget to APPLY the CSS code and SAVE the chat window.