Set authentication cookies to be shorter but then extend with every page load

You do not need to rewrite wp_set_auth_cookie(), it allows you to change the expiration time of the cookie:

add_filter( 'auth_cookie_expiration', 'wpse101378_change_expire_time', 3 );
function wpse101378_change_expire_time( $expire, $user_id, $remember ){
    //The $remember variable indicates whether the user has elected
    //to be 'remembered'.
    //By default, if true, WP sets expire to 14 days if false, 2 days

    //$expire is time in seconds

    return 24*60*60;
}

If you want to clear the current cookie, you can use wp_clear_auth_cookie() (see source)