Spit out list of distinct instances of custom field?

Here’s a dashboard widget you could manage them with:

// add dashboard widget
add_action('wp_dashboard_setup', 'wpse_dashboard_widget');
function wpse_dashboard_widget() {
    // only show widget to users who can publish pages
    if(current_user_can('publish_pages')) {
        wp_add_dashboard_widget('deduped_dash_widget', 'Publications', 'wpse_create_deduped_dash_widget');
    }
}
// widget contents
function wpse_create_deduped_dash_widget() {
    // select publications without duplicates by using DISTINCT
    global $wpdb;
    $mentions = $wpdb->get_results("SELECT DISTINCT meta_value FROM wp_postmeta WHERE meta_key = 'press_mention_information_publication-name'");
    // if any were found
    if($mentions) {
        // display them in an unordered list; could do <ol> if you want numbers
        echo '<ul>';
        foreach($mentions as $publication) {
            echo '<li>' . $publication->meta_value) . '</li>';
        }
        echo '</ul>';
    }
}

If you have a custom theme or custom child theme you can place this in functions.php. Otherwise, you can create a plugin which may be easier than creating a child theme just for this widget.

Or, if you have access to phpMyAdmin, you could run the query

SELECT DISTINCT meta_value FROM wp_postmeta WHERE meta_key = 'press_mention_information_publication-name'

and get the list as well.