How can I add new attributes in a Class when my addon is active?

You need to declare a location for this data in your class, like an array beforehand and then use it like below:

class test {
  private $first_name;
  private $data = array();

  public function __set($name, $value) {
    $this->data[$name] = $value;
  }

  public function __get($name) {
    if (array_key_exists($name, $this->data))
        return $this->data[$name];
    else
        return 'not found';
  }
}

$try = new test();
$try->last_name="jhon";

echo $try->last_name;

Then look at the register_activation_hook from WordPress.

You can also read more about Magic Methods on the PHP website: http://php.net/manual/ro/language.oop5.overloading.php#object.set