The Kharijites had plans to assasinate Caliph Ali, Muawiyah and Amr ibn Al-Aas who at that time ruled Egypt. These three men were the most powerful in the Rashidun Caliphate at that time. With Ali and Muawiyah both killed the First Fitna ends prematurely, and the Ummayad Caliphate will never exist. I really can't see Muawiyah's son Yazid commanding the same amount of respect. Amr ibn al-Aas was too old to stake any sort of claim on the entire realm.
I can see Ali's son Hasan being chosen as Caliph. If not him perhaps Talha or Zubayr. It was Muawiyah who changed the Caliphate into a hereditary office. Without him the Caliph will continue to be elected and the most likely choices are remaining companions of Muhammad.
Ali and Muawiyah agreed that after Ali's death Muawiyah would rule, and then the Caliphate would transfer to Ali's son Hasan. If there is no Muawiyah I can see Hasan taking the Caliphate without contest. Though if this also devolves into a hereditary office I am not sure.
This is honestly a huge point of divergence that can have massive implications on the world, and one that sadly isn't very explored!