you can wrap your function and hook it to ajax call like this:

//if you want only logged in users to access this function use this hook
add_action('wp_ajax_more_links', 'my_AJAX_more_links_function');
//if you want none logged in users to access this function use this hook
add_action('wp_ajax_nopriv_more_links', 'my_AJAX_more_links_function');

function my_AJAX_more_links_function(){
    check_ajax_referer('more_links');
    $success_response = new WP_Ajax_Response();
    $args = array('posts_per_page' => 20 , 'offset' => $_POST['offset']);
    $sidebar = new WP_Query($args);
    if ( $sidebar->have_posts() ){ 
         while ( $sidebar->have_posts() ) {
            $sidebar->the_post(); 
            $out .= '<div class="story">';
            $out .= '<a href="' . the_permalink().'" title="'. the_title'.">' . the_title().' - '.the_time("F j, Y h:i A") .'</a></div>';
        }
        $success_response->add(array(
                    'what' => 'has',
                    'data' => array('html' => $out, 'offset' => $_POST['offset']
        ));
    }else{
        $out = __('Sorry but No more!');
        $success_response->add(array(
                    'what' => 'none',
                    'data' => $out
                ));
    }

    $success_response->send();      
    exit;   
}

then add this to your sidebar function at the end

<span class="more_links"></span>
<span class="get_more">
    <input type="hidden" name="offset" id="offset" value="20">
    <input type="submit" name="more" id="more" value="Get more links">
</span>
<script type="text/javascript" >
jQuery(document).ready(function($) {
    jQuery('#more').click(function() { 
        var data = {
            action: 'more_links',
            offset: jQuery( '#offset' ).val(),
            _ajax_nonce: <?php echo wp_create_nonce( 'more_links' ); ?>
        };

        // since 2.8 ajaxurl is always defined in the admin header and points to admin-ajax.php
        jQuery.post(ajaxurl, data, function(response) {
            var res = wpAjax.parseAjaxResponse(response, 'ajax-response');
            jQuery.each( res.responses, function() { 
                if (this.what == 'has') {
                    //insert links
                    jQuery(".more_links").append( this.data.html ); 
                    //update offset value
                    jQuery("#offset").val(this.data.offset);
                    jQuery(".more_links").fadeIn("fast");
                }else{
                    //no more links found
                    jQuery(".more_links").append( this.data.html );
                    jQuery(".get_more").remove();

                }
                //end if
                });//end each


        });

        return false;
    })
});
</script>

and there you go, oh wait you need to add wp-ajax-response so

add_action('wp_head','add_scripts_121');
function add_scripts_121(){
    wp_enqueue_script('wp-ajax-response');
}

and you are set