How can I make uploaded images in the editor load with HTTPS?

This should do it:

add_action('admin_notices', 'https_the_content');
function https_the_content() {

    global $post;

    if(!$post->post_content)
        return;

    //change src to use the current url scheme
    $post->post_content = str_replace(array("src=\"http://", "src="http://"), array("src=\"//", "src="https://"), $post->post_content);   

    return $post->post_content;

}