Add to basic site settings a custom field

For example, you have to add a custom field to "Basic site settings" page (/admin/config/system/site-information).

Let's say it is a phone number of your site.

For this new field you have to implement hook hook_config_schema_info_alter() form in *.module file.

<?php

use Drupal\Core\Form\FormStateInterface;

/**
 * Implements hook_config_schema_info_alter().
 */
function MODULENAME_config_schema_info_alter(array &$definitions): void {
  $definitions['system.site']['mapping']['phone'] = [
    'type' => 'string',
    'label' => t('Site phone'),
  ];
}

/**
 * Implements hook_form_system_site_information_settings_alter().
 */
function MODULENAME_form_system_site_information_settings_alter(array &$form, FormStateInterface $form_state): void {
  $form['site_information']['phone'] = [
    '#type' => 'textfield',
    '#title' => t('Phone'),
    '#default_value' => \Drupal::config('system.site')->get('phone'),
  ];

  $form['#submit'][] = 'MODULENAME_form_system_site_information_settings_submit';
}

/**
 * Implements hook_form_system_site_information_settings_submit().
 */
function MODULENAME_form_system_site_information_settings_submit(array &$form, FormStateInterface $form_state): void {
  \Drupal::configFactory()->getEditable('system.site')
    ->set('phone', $form_state->getValue('phone'))
    ->save();
}

Also, it can be helpful create a token for this phone field in the same *.module file.

<?php

/**
 * Implements hook_token_info().
 */
function MODULENAME_token_info(): array {
  $info['tokens']['site']['phone'] = [
    'name' => t('Phone'),
    'description' => t('Site phone.'),
  ];

  return $info;
}

/**
 * Implements hook_tokens().
 */
function MODULENAME_tokens(string $type, array $tokens): array {
  $replacements = [];

  if ($type === 'site') {
    foreach ($tokens as $name => $original) {
      if ($name === 'phone') {
        $replacements[$original] = \Drupal::config('system.site')->get('phone');
      }
    }
  }

  return $replacements;
}

So, this is it.