If Tyrion’s mother had not died during childbirth.
1. One main reason in the Game of thrones why Tywin hates Tyrion is because Jaime is a King’s guard and cannot become his Heir, hence Tywin is left with Tyrion as his heir. Had Tyrion been a dwarf and had Joanna not been dead, Tywin would have had been different with his son. Had Joanna lived, Jaime and Cersei would have been married off to Elia and Oberyn respectively. Cersei would have been sent off to Drone. Elia would not have been raped and killed by the Mountain (Ser Gregor Clegane). Jaime would have married Elia and stayed at Casterly Rock, not becoming a King’s guard, thus becoming heir after Tywin.
This would have made the relation between Tywin and Tyrion much better. But the incident with Tysha shamed Tywin so much and after that incident, Tyrion did not do anything that spoilt the Lannister name. Had Joanna been alive, she would not have let Tywin treat Tyrion the way he did. She would have always been with Tyrion to support him, like Jaime. Cersei would not have anything against him.
2. Tyrion was a dwarf and not a good looking one at that. But he always was the good sort among the Lannisters and he read a lot too. Tyrion is always the one ready with ideas that prove to work out in situations of great distress, and this would have made Tywin like Tyrion more.
Had the Lannister twins been married over to the Dornish heirs, the Starks would have married the Targaryens and Robert’s rebellion would never have happened. Hence, it would still have been the rule of the Targaryens. Daenerys would have never become the queen she is now nor would she have gotten her Dragons, instead her brother Rhaegar would have been the king with Lyanna as his queen. Jaime would not be there to kill the Mad King, hence he would have burned everything until someone would have stopped him by killing him. This would naturally put Rhaegar on the Iron Throne. He was not mad like his father and would have ruled wisely.
He loved Lyanna and would have always kept her happy. Jon Snow would not have been a bastard but a King of the Iron throne. He would not have gone to the wall nor would have met Ygritte or have become Lord Commander.
3. In a matter of events where Tywin did not hate Tyrion, things would have been a lot different for the Starks. The Lannisters would have married the Dornish Heirs and Starks the Targaryens. Hence, Catelyn would have been married to Brandon Stark as she was betrothed and none of the Stark children in the series would have existed. Jon Arryn would not have been Hand of the King and would have ruled over the Vale. The rebellion against the Mad King would not have happened. This would ensure the Targaryen rule in Westeros. The Khal tribe would still be afraid of the Narrow sea and would dare not cross it. Jon would not have gone to the wall and would not have fought the white walkers or befriended the Wildings. Ramsay would not have gone to Winterfell nor would Thoen have been castrated. He would have still been ogling girls.
Had Joanna been alive Tywin would have been a less bitter man and would not be as harsh towards Tyrion as he is now. Everything in the seven Kingdoms would have been completely a different scenario with Joanna in it.