How do I add settings to the Background Options Page?

The page content for custom backgrounds is created in wp-admin/custom-background.php. There is no action available where the fields are printed.

To add new fields we have to print JavaScript into the page footer. The action is 'admin_footer-appearance_page_custom-background'.

To save those field values we have to hook into 'load-appearance_page_custom-background'.

The following example adds a new option for background-origin – useful if you have set a border around body in your stylesheet.

add_action(
    'load-appearance_page_custom-background',
    array ( 'T5_Background_Origin', 'get_instance' )
);

/**
 * Add a new row to the background options table for 'background-origin'.
 *
 * @author  Thomas Scholz http://toscho.de
 * @version 2012.09.10
 */
class T5_Background_Origin
{
    /**
     * Main instance.
     * @type object|NULL
     */
    protected static $instance = NULL;

    /**
     * The name for the option. Will be saved as theme option.
     *
     * @link http://www.w3.org/TR/css3-background/#the-background-origin
     * @type string
     */
    protected $option = 'background_origin';

    /**
     * Label on the left side of our new option.
     *
     * @type string
     */
    protected $table_header="Background Origin";

    /**
     * Return an instance.
     *
     * @wp-hook load-appearance_page_custom-background
     * @return object
     */
    public static function get_instance()
    {
        NULL === self::$instance and self::$instance = new self;
        return self::$instance;
    }

    /**
     * Save our option and register the form.
     *
     * @wp-hook load-appearance_page_custom-background
     */
    public function __construct()
    {
        add_action(
            'admin_footer-appearance_page_custom-background',
            array ( $this, 'form' )
        );

        if ( empty ( $_POST[ $this->option ] ) )
        {
            return;
        }

        check_admin_referer( $this->option, "_t5nonce-$this->option" );
        set_theme_mod( $this->option, $_POST[ $this->option ] );
    }

    /**
     * Create the form elements.
     *
     * @wp-hook admin_footer-appearance_page_custom-background
     * @return void
     */
    public function form()
    {
        $nonce = wp_nonce_field(
            $this->option,
            "_t5nonce-$this->option",
            TRUE, // check referer
            FALSE // do not echo
            );
        $html = $nonce . $this->get_radio_fields();
        $this->print_script( $html );
    }

    /**
     * Create the jQuery function that inserts our form fields.
     *
     * @param  string $html Radio buttons
     * @return void
     */
    protected function print_script( $html )
    {
        $row = "'<tr><th>$this->table_header</th><td>$html</td></tr>'";
        ?>
<script>jQuery( function <?php echo $this->option; ?>($) {
    $('.form-table:last').append(<?php echo $row; ?>);
});</script>
<?php
    }

    /**
     * Helper for form(). Create radio input fields
     *
     * @return string
     */
    protected function get_radio_fields()
    {
        $value  = get_theme_mod( $this->option, 'padding-box' );
        $radios = array ( 'padding-box', 'border-box', 'content-box' );
        $html   = "<p>";

        foreach ( $radios as $radio )
        {
            $html .= sprintf(
                ' <input type="radio" name="%1$s" value="%2$s" id="%3$s"%4$s>'
                . ' <label for="%3$s">%2$s</label> ',
                $this->option,
                $radio,
                "$this->option-$radio",
                // returns ' as value delimiters and has to be escaped
                addslashes( checked( $value, $radio, FALSE ) )
            );
        }

        return "$html</p>";
    }
}