Can destiny change if a young man prays to his Lord to marry a girl he loves, knowing that her family rejects him because he currently cannot afford the costs of marriage, and they are trying to marry her to another?

There is no harm in supplicating that Allah make a girl your destiny as long as she is not married. If she is married, then it is not permissible to supplicate for that, because it is a supplication for sin. And supplication repels divine decree by the permission of Allah Almighty.