From the Shi'a point of view an Imam is not chosen by people. Even the ma'soom Imam himself does not chose who the next Imam will be. It is something exclusively appointed by Almighty God.
We clearly know that an Imam must have certain characteristics, the most important of them is 'Ismah, which is not something that can be acquired, and it is something endowed by the Almighty.
In addition to this, we see that the Prophet had mentioned who the Ahlul Bayt (a.s.) were, and the numerous famous traditions of the Twelve Khalifahs have also mentioned this.
Even in the case of Imam Husain (a.s.), it was one of his sons who was divinely selected and appointed to be the next Imam.
Another point that can be mentioned here is that the descendents of Imam Hasan (a.s.) did have a continuous role in Imamah.One example for this is that Imam Muhammad al-Baqir's (a.s.) mother was the daughter of Imam al-Mujtaba (a.s.). This means that Imam al-Baqir (a.s.) goes back to both Imam Hasan and Imam Husain (a.s.).
There is a tradition that says as a result of the greatest sacrifice made by Imam Husain (a.s.), he was blessed with three things:
1. The lineage of the Imams are from him.
2. There is cure in his soil.
3. Prayers are answered under his dome.
May the Almighty grant us their shafa'ah.