Quick & simple: You could simply use post__in in your query with the post IDs of those in your pool, though it might be less easy to maintain when you add and remove posts.
Or you could use a meta field and change your query to include it, see the WP_Query documentation. Then either set that field manually in the post editor or use a Plugin (Advanced Custom Fields, Custom Field Suite etc pp) to generate a fancy looking checkbox.
To use the custom field version, change your get_posts() call from
$rand_posts = get_posts('numberposts=3&orderby=rand&post_type=any&cat=-15');
to
$rand_posts = get_posts('numberposts=3&orderby=rand&post_type=any&cat=-15&meta_key=toppost&meta_value=1');
and you should be up and running, assuming the meta field you’re adding is called toppost and you’ve set its value to 1 for the top posts you want in your pool.