ExtendedSiteInformationForm.php

<?php

// in src/Form directory.

namespace Drupal\mymodule\Form;

use Drupal\Core\Form\FormStateInterface;
use Drupal\system\Form\SiteInformationForm;

/**
 * Extends drupal core SiteInformationForm, Add a new text field to save api
 * key for our site.
 *
 * Class ExtendedSiteInformationForm
 *
 * @package Drupal\api_settings\Form
 */
class ExtendedSiteInformationForm extends SiteInformationForm {

  /**
   * {@inheritdoc}
   */
  public function buildForm(array $form, FormStateInterface $form_state) {
    // Build parent form.
    $form = parent::buildForm($form, $form_state);

    // Add siteapikey text box to site information group.
    $form['new_element'] = [
      '#type' => 'textfield',
      // ... more attributes
    ];

    return $form;
  }
}

mymodule.services.yml

services:
  mymodule.route_subscriber:
    class: Drupal\mymodule\Routing\RouteSubscriber
    tags:
      - { name: event_subscriber }

RouteSubscriber.php

<?php

// in src/Routing directory.

namespace Drupal\mymodule\Routing;

use Drupal\Core\Routing\RouteSubscriberBase;
use Symfony\Component\Routing\RouteCollection;

/**
 * Provides a routing subscriber which change _form key of
 * "system.site_information_settings" route to override that form. Class
 *
 * Class RouteSubscriber
 *
 * @package Drupal\api_settings\Routing
 */
class RouteSubscriber extends RouteSubscriberBase {

  /**
   * {@inheritdoc}
   */
  protected function alterRoutes(RouteCollection $collection) {
    if ($route = $collection->get('system.site_information_settings')) {
      $route->setDefault('_form', '\Drupal\mymodule\Form\ExtendedSiteInformationForm');
    }
  }

}