Best way to add some custom javascript using jquery to a child theme

Here you go.

add_action('init', 'register_custom_jquery');
function register_custom_jquery() {
    wp_deregister_script('jquery');
    wp_register_script(
        'jquery', 
        'https://ajax.googleapis.com/ajax/libs/jquery/1.6.1/jquery.min.js', 
        array(), 
        '1.6.1'
    );

    wp_register_script(
        'my-custom-script', 
        'http://example.com/js/script.js', 
        array(), 
        '1.0'
    );
}

add_action('wp_enqueue_scripts', 'add_js_to_page');
function add_js_to_page() {
    wp_enqueue_script( 'jquery' );

    wp_enqueue_script( 'my-custom-script' );
}