Adding new settings to existing options is straightforward, but can require looking into the WordPress core code to find out which hooks to utilise.

The following example adds an "extra blog description" field.

/**
 * Class for adding a new field to the options-general.php page
 */
class Add_Settings_Field {

	/**
	 * Class constructor
	 */
	public function __construct() {
		add_action( 'admin_init' , array( $this , 'register_fields' ) );
	}

	/**
	 * Add new fields to wp-admin/options-general.php page
	 */
	public function register_fields() {
		register_setting( 'general', 'extra_blog_description', 'esc_attr' );
		add_settings_field(
			'extra_blog_desc_id',
			'<label for="extra_blog_desc_id">' . __( 'Blog description' , 'extra_blog_description' ) . '</label>',
			array( $this, 'fields_html' ),
			'general'
		);
	}

	/**
	 * HTML for extra settings
	 */
	public function fields_html() {
		$value = get_option( 'extra_blog_description', '' );
		echo '<input type="text" id="extra_blog_desc_id" name="extra_blog_description" value="' . esc_attr( $value ) . '" />';
	}

}
new Add_Settings_Field();

Another explanation can be found at http://trepmal.com/2011/03/07/add-field-to-general-settings-page/