How to show a custom meta box on the “Quick Edit” screen?

There seems to be no easy way to do this, you must add all code yourself. inline_edit_row(), the function that draws the Quick Edit and Bulk Edit screens, seems to have only one action that you can hook into: quick_edit_custom_box or bulk_edit_custom_box. It gets called for all non-core columns that wp_manage_posts_columns() returns. There are some filters you can use to add a column, for example manage_posts_columns. Unfortunately, this function defines the column headers of the post table, so you should remove it again before print_column_headers() prints them. This can be done in the get_column_headers() function, with the manage_[screen_id]_headers filter. edit-post is the screen id for the Edit Posts screen.

All together, this gives a hack like the following to add some code. Finding out where you can handle the form submission is (currently) left as a exercise to the reader.

// Add a dummy column for the `posts` post type    
add_filter('manage_posts_columns', 'add_dummy_column', 10, 2);
function add_dummy_column($posts_columns, $post_type)
{
    $posts_columns['dummy'] = 'Dummy column';
    return $posts_columns;
}
// But remove it again on the edit screen (other screens to?)
add_filter('manage_edit-post_columns', 'remove_dummy_column');
function remove_dummy_column($posts_columns)
{
    unset($posts_columns['dummy']);
    return $posts_columns;
}

// Add our text to the quick edit box
add_action('quick_edit_custom_box', 'on_quick_edit_custom_box', 10, 2);
function on_quick_edit_custom_box($column_name, $post_type)
{
    if ('dummy' == $column_name) {
        echo 'Extra content in the quick edit box';
    }
}

// Add our text to the bulk edit box
add_action('bulk_edit_custom_box', 'on_bulk_edit_custom_box', 10, 2);
function on_bulk_edit_custom_box($column_name, $post_type)
{
    if ('dummy' == $column_name) {
        echo 'Extra content in the bulk edit box';
    }
}