Bismihi ta'ala

We believe that the names, titles and kunyahs (teknonym) of our Ma'sumeen (a.s.) and certain eminent members of their family are chosen by Allah ta'ala. It was also a custom among Arabs at that time to give a kunyah to a person who didnt have a child with that name. 

Imam Muhammad al-Jawad (a.s.) has kunyah of Abu Ja'far, and he did not have a child by the name of Abu Ja'far. 

Imam al-Mahdi (a.a.f.) has the kunyah Aba Saleh, and he does not have children.

Imam al-Sadiq (a.s.) has kunyah of Abu Abdillah, but his elder son was Isma'il. Although he did have a son by the name Abdullah (al-Aftah), but it would not be possible for the Imam to have that kunyah for that reason, because Abdullah al-Aftah was deviant. 

As for Imam Husain (a.s.), his kunyah was also Abu Abdillah, and it is not clear exactly why, but it has been explicitly mentioned from the Messenger of God (s.a.w.). One narration says that the Prophet (s.a.w.) used this kunyah with Imam Husain (a.s.) when he was at the age of two. 

Therefore, his kunyah has been transmitted as such, from the Prophet (s.a.w.), and of course what the Prophet does is from Allah ta'ala.

And Allah knows best