The sons of Imam Husayn came to be noticed not because of Imam Husayn’s campaign against tyranny but due to their own ability and capacity. In knowledge, in practice, in endurance and attitude they stood perfect, each in his time and age. No other one than they were more befitted to the leadership. The nine sons, one after the other, from Zainul Abideen to the twelfth one Mahdi became Imams due to their own deservation.
Of course the sacrifice done by Imam Husayn and his martyrdom at Karbala enhanced the popularity of his sons. God too as a reward to his martyrdom settled the Imam-hood in his progeny. As the Quranic Verse says:
“Indeed, God purged Adam and Noah and the progeny of Abraham and the house of Imran over the worlds;" (3:33)
It can be deduced that the same might have been the Divine intention with regards the turn of Imam Husayn;
“God knows as how and where to house His Mission.”