Bismihi ta'ala

As we know, having a kunya, or teknonym is mustahab, and customarily it is saying the father/mother of, or Abū/Um , and the name of the elder son. At times it could be the elder daughter, or maybe not the elder son. Of course, this is also something very special and unique to the Arabic language as well. 

For example, for the holy Prophet (s.a.w.), we say Abū al-Qāsim, or Imam al-Sadiq (a.s.), we say Abū Abdullah, even though his elder son was Isma'il. 

Or for Imam al-Mahdi (a.a.f.), we say Abā Sālih. And other Imams (a.s.) as well. 

Even with Abū Sufyān, apparently in history, he did not have a son by the name of Sufyān. 

Although it is said the Imam Husain (a.s.) did have a son by the name of 'Abdullah, but he was not the older son.

As Shi'ah, our belief is that the names, the titles, and the teknonyms of our Ma'sumeen (a.s.) are all designated and appointed by Allah ta'ala, through Gabriel (a.s.) and given by the Prophet (s.a.w.). 

There has been mentioning that the holy Prophet (s.a.w.) referred to Imam Husain (a.s.) with this kunya while he was still very young. 

And Allah knows best

اللهمّ بالحسين الوجيه وجدّه وأبيه واُمه وأخيه والمعصومين من بنيه اجعلني مع الحسين وآله الطاهرين وأصحابه الطيّبين .