Undefined array key "name" in system_requirements() (line 34 di /var/www/html/web/modules/system/system.install

Thanks for reporting this. This warning is not present on the clean D7 installation. Looking at the code, I suppose that the site could be using an installation profile which is not available anymore and therefore system_get_info('module', $profile) would return an empty array. See:

    // Display the currently active installation profile, if the site
    // is not running the default installation profile.
    $profile = drupal_get_profile();
    if ($profile != 'standard') {
      $info = system_get_info('module', $profile);
      $requirements['install_profile'] = array(
        'title' => $t('Install profile'),
        'value' => $t('%profile_name (%profile-%version)', array(
          '%profile_name' => $info['name'],
          '%profile' => $profile,
          '%version' => $info['version']
        )),
        'severity' => REQUIREMENT_INFO,
        'weight' => -9
      );
    }

Can you please provide more information? Thanks!