CSS is the way to go here.
Add a new CSS for admin using the admin_enqueue_scripts hook in your child theme functions.php (if you don’t have one, you’re doing something wrong)
function my_admin_enqueue_style() {
wp_enqueue_style('my-css', get_stylesheet_directory_uri().'/css/my-admin.css', array(), '1.0.0');
}
add_action('admin_enqueue_scripts', 'my_admin_enqueue_style');
In your my-admin.css file
/* change the content to another Dashicon */
#wp-content-editor-tools .wp-media-buttons .add_media span.wp-media-buttons-icon::before {
content: "\f104";
}
/* or set a background-image instead (set content to "" in the rule before) */
#wp-content-editor-tools .wp-media-buttons .add_media span.wp-media-buttons-icon {
background-image: url(../../wp-content/uploads/2017/10/my-image.png);
background-position: center;
}