Hi,
to be honest the best way would be to have custom module with javascript that allows user select color and that color code will be used for < body > - this would be option if guest/users have to choose colors.
If you want to decide about bg color you can play with class names that will override body bg color and use that class name inside menu item settings. Or as alternative method use CustomCSS module with css to change colors published in debug position and assigned to different subpages using a rules (you can use Advanced Module Manager from RegularLabs for advanced rules).
That's all I can suggest for free here, because this seems to be customization question. But you got solid tips already.