How can I pick a single post from the latest 3?

Here’s my approach…

First you have to select 3 latest posts, then you have to pick random one of them…

But it’s easier to shuffle selected posts than picking only one of them – that way you can still use normal loop:

<?php
    $args = array(
        'post_type' => 'post',
        'posts_per_page' => 3,
        'post_status'   => 'publish'
    );    
    $rand_query = new WP_Query( $args );

    shuffle( $rand_query->posts );

    if ( $rand_query->have_posts() ) :
        while ( $rand_query->have_posts() ) : $rand_query->the_post();
?>

    // HERE GOES THE DIV WITH POST

<?php
            break;  // we want only one post to be shown, so we break the loop
        endwhile;
    endif;
?>