This field formatter can be used on entity reference fields to select a random entity from the list to render. Note that the more items you have, the better the 'random' pick will seem. Also note that this will only fire once per cache write, then again on cache clear.

RandomEntityFieldFormatter.php

<?php

namespace Drupal\harlib_random_entity_display\Plugin\Field\FieldFormatter;

use Drupal\Core\Field\FieldItemListInterface;
use Drupal\Core\Field\Plugin\Field\FieldFormatter\EntityReferenceEntityFormatter;

/**
 * Plugin implementation of the 'random_entity_field_formatter' formatter.
 *
 * @FieldFormatter(
 *   id = "random_entity_field_formatter",
 *   label = @Translation("Random rendered entity"),
 *   field_types = {
 *     "entity_reference"
 *   }
 * )
 */
final class RandomEntityFieldFormatter extends EntityReferenceEntityFormatter {

  public function settingsSummary() {
    $summary = [];

    $view_modes = $this->entityDisplayRepository->getViewModeOptions($this->getFieldSetting('target_type'));
    $view_mode = $this->getSetting('view_mode');
    $summary[] = $this->t('Randomly selected entity will be rendered.');
    $summary[] = $this->t('Rendered as @mode', ['@mode' => isset($view_modes[$view_mode]) ? $view_modes[$view_mode] : $view_mode]);

    return $summary;
  }

  /**
   * {@inheritdoc}
   */
  public function viewElements(FieldItemListInterface $items, $langcode) {
    $view_mode = $this->getSetting('view_mode');
    $elements = [];

    $all = $this->getEntitiesToView($items, $langcode);

    if (count($all)) {
      $entity = $all[array_rand($all, 1)];

      $view_builder = $this->entityTypeManager->getViewBuilder($entity->getEntityTypeId());
      $elements[] = $view_builder->view($entity, $view_mode, $entity->language()
        ->getId());
    }

    return $elements;
  }

}