My niece has been divorced three times - I understand that she cannot return back to her husband till she gets married to another man. But her husband tells her that because one of the Talaaq did not pass the Iddah time, that is, three months. He considers it as two Talaaqs only. Does each Talaaq have to pass the three months Iddah to be considered as Talaaq? In other words, does the time gap matter? Example, if he divorced her today and returns back to her, then in a week's time he divorce her again, is it considered as two divorces or one divorce?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Since the husband of your niece has divorced her three times, then this means that such a divorce is final and she becomes forbidden to him as a wife unless she marries another man [a serious marriage and he consummates the marriage with her and then he divorces her or dies]. In simple words, you have to prevent such a husband from having any relation with her. Then she can marry another man while fulfilling all conditions of marriage and without any form of deception. When this latter husband divorces her normally, then she can return to the previous husband but with a new contract of marriage.
As for the husband's allegation that one of the divorces took place before completing her 'Iddah from the previous divorce, then this has no effect on the next divorce.

Allah knows best.