How do I create a random post that will last for a day

Here is some code doing what you’ve asked and using the ideas others have highlighted:

<?php
if ( false === ( $totd_trans_post_id = get_transient( 'totd_trans_post_id' ) ) ) {
     $args = array('numberposts' => 1, 'orderby' => 'rand');
     $totd = get_posts($args);
     $midnight = strtotime('midnight +1 day');
     $timenow = time();
     $timetillmidnight = $midnight - $timenow;
     echo $midnight;
     echo ",".$timenow;
     set_transient('totd_trans_post_id', $totd[0]->ID, $timetillmidnight);
} else {
    $args = array('post__in' => array($totd_trans_post_id));
    $totd = get_posts($args);
}

foreach( $totd as $post ) : setup_postdata($post); ?>
    <div>
        <a href="https://wordpress.stackexchange.com/questions/17056/<?php the_permalink(); ?>"><?php the_title(); ?></a>
        <?php the_content(); ?>
    </div>
<?php endforeach; ?>

get_posts() help Plus the way I have coded it should make the Tip of the Day post change at midnight every day.

This can be improved upon because we are showing random there is nothing stopping the same post showing twice in a row…