Ismail son of Imam Jaffar as-Sadiq

This answer was provided on ShiaChat a while ago (Source)

There are some narrations that claim Imam Al-Sadiq said that Bada' occurred after his son Ismail died, but our scholars state that these traditions are not referring to the Imamate of Ismail. Instead, Ismail was supposed to die, but Imam Al-Sadiq (as) asked Allah (swt) to keep him alive, and Allah (swt) did keep him alive. So Bada' occurred here and he remained alive the first time, but he later died. Some took such narrations to indicate that Bada' occurred in Imamate, but it wasn't so, as Imam Al-Sadeq (as) was referring to the expected death of his son Ismail.

Imam Al-Sadeq (as) greatly loved his son Ismail, and he was very dear to his heart. He was also his oldest son. Thus, many during the life of Imam Al-Sadeq (as) thought that Ismail would be the next Imam. When he died, however, it became very clear to everyone that he could not have been the Imam. For many reasons, many of which were political, Imam Al-Sadeq (as) did not publicly announce the Imamate of Al-Kadhem (as). He had only informed those whom he trusted and were firm believers in Allah (swt).

Ismail died in 142 A.H., and Imam Al-Kadhem (as) was born in 128 A.H., which means that Imam Al-Kadhem (as) was 14 years old when his brother Ismail died. The nass on Imam Al-Kadhem's Imamate was not only given when he was in the cradle or five years old. During various stages of Imam Al-Kadhem's (as) life, Imam Al-Sadeq (as) declared him as the next Imam.