Why, Among the Daughters of the beloved Prophet of Islam (PBUH), only Hazrat Fatima (A.S) was Infallible?

       According to the principles and foundations of the Qur'an and hadiths; The selection of the Prophet, Imam and their introducer to the people, as well as the introducer of the infallible, is done by God Almighty:

    When a sign came to them they said: 'We will not believe in it unless we are given that which the Messengers of Allah have been given' But Allah knows best where to place His Message..."[1]  

    Because only God knows who is worthy of this position and others are useless from this knowledge. For this reason, the introduction of the innocent must be done by God and through the prophet or her successor.

      Therefore; someone's innocence is not related and conditional on whether that person is the son of the Prophet or not;

        As the children of all prophets were not infallible, there is no requirement that all children of a prophet are infallible.

In the meantime, we can mention the children of Hazrat Yaqub, out of his twelve children, only Hazrat Yusuf was infallible and a prophet and his other children were not only infallible, but some of them were close to contaminating their hands with the pure blood of Yusuf (A.S).

Read the following links for more information:

"Meaning of innocence", question 1738

"Innocent examples", question 1749

"The exclusiveness of the innocent in fourteen people", question 178
 

Reference:

[1] [1]. Ana'am, 124.