How to change the Header Right sidebar registration parameters

header-right.php

add_action( 'register_sidebar', 'change_header_right_registration', 9999 );
/**
 * Change the Header Right Sidebar registration parameters.
 * 
 * To change the header right (or any) sidebar registration parameter, you'll
 * need to use the global after the sidebar is registered.  Within the function
 * `register_sidebar()` in WordPress Core, there is an action event after
 * the sidebar is registered.  We register to that event and then change
 * each parameter in the global `$$wp_registered_sidebars`.
 *
 * @since 1.0.0
 *
 * @param array $args Sidebar registration arguments
 *
 * @return void
 */
function change_header_right_registration( array $args ) {
	if ( $args['id'] !== 'header-right' ) {
		return;
	}
	global $wp_registered_sidebars;

	$wp_registered_sidebars[ $args['id'] ]['name'] = 'Page Name';
	$wp_registered_sidebars[ $args['id'] ]['before_title'] = str_replace( '<h3', '<h1', $wp_registered_sidebars[ $args['id'] ]['before_title'] );
	$wp_registered_sidebars[ $args['id'] ]['after_title'] = str_replace( '</h3>', '</h1>', $wp_registered_sidebars[ $args['id'] ]['after_title'] );
}