Code

<?php

    $parent = get_cat_ID( 'photos' );
    $cats = get_categories("child_of=".$parent);

    foreach ($cats as $cat) {
    query_posts( array (
        'posts_per_page' => 1,
        'orderby' => 'rand',
        'cat' => $cat->cat_ID
    ) );
?>
<?php if ( have_posts() ) : while ( have_posts() ) : the_post(); ?>
    <ul>
        <?php if (has_post_thumbnail()) { ?>
        <?php
            printf( '<li><a href="'.get_category_link($cat->cat_ID).'">'.get_the_post_thumbnail(get_the_ID(), "album-thumb").'</a><a href="'.get_category_link($cat->term_id).'">%s</a></li>',
            apply_filters('get_term', $cat->name));
        ?>
        <?php } ?>
    </ul>
<?php endwhile; endif; } ?>
    <?php if (!$cat == true) { ?>
        <p><?php _e("Sorry, no posts matched your criteria."); ?></p>
    <?php } ?>
<?php wp_reset_query(); ?>