Different header background for mobile

You can use css to change the header image if you want to be able to set it at a specific screen resolution. (Use on a child theme style.css)

Ex. For screen sizes 840px and below.

@media screen and (max-width:840px){
   .body #header {
      background-image: url('http://example.com/image.jpg');
   }
}

You can also make use of when WordPress detects that it is being viewed on a mobile device. (Function to be used on a child theme functions.php)

add_filter( 'theme_mod_header_image', 'custom_mobile_header_image' );

function custom_mobile_header_image( $url ) {
   if ( wp_is_mobile() ) {
      $url = 'http://example.com/image.jpg';
   }
   return $url;
}