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In another world, Lily would have stayed home after the argument. James would have left to visit Sirius and raged at him for a while, before realising that he was being an asshole and returning home, where he would have fantastic make-up sex with his wife.
In this world, however, Lily left. Her Aunt Rose had a gorgeous piano and Lily loved playing it. Now that she was out of Hogwarts, she could play it all year, instead of just during the holidays. Her parents had teased her about returning home for it, rather than for them. Petunia had just huffed, and left for her date with her incredibly boring boyfriend.
Lily had always been close to her Aunt Rose. When she started to make things float, or change Petunia's hair purple after she had said something mean, it was Aunt Rose she went to. And Aunt Rose had told her of other people with special powers, of secret societies with the ability to create illusions or travel through time. When her Hogwarts letter came, it was as shocking to Aunt Rose as it was to Lily. Aunt Rose had told her that she must never, ever mention the stories to anyone else. To Lily's dismay, she also stopped telling her the stories. If Lily wasn't flame-active, if it was a different power that she had, the less she knew the better.
The evening that Lily stormed out of her house in a tearful rage, Aunt Rose had a guest. Stefano Gokudera, an Italian Mafia boss and a distant cousin, was visiting her while he was in England on business. He heard the music start and was instantly entranced. Looking into the drawing room, he saw the most beautiful young woman he'd ever seen. She looked like an angel, sent to create heavenly music. Lily was flattered by the attention of the older, handsome man. He was interested in her opinions about music, the books she'd read, her childhood. He shared stories of Aunt Rose and of his home in Italy. She knew that he was leaving England the next day, and in her anger at her husband, she saw no problem with accepting his invitation, and spending the night with someone who would appreciate her.
Nine months later, Lily gave birth to a baby boy. James knew immediately that the child couldn't be his. He had his mother's bright green eyes, but the shock of white hair couldn't have come from either of them. Lily confessed her infidelity with tears in her eyes. She hadn't known for sure whether the child was Stefano's, so she had kept silent and hoped. In the desperate chaos of war, there had been no time for confessions of infidelity. There had barely been time for them to spend together at all.
James had matured since Hogwarts. He knew that the pressure of the war had affected them both, and he truly loved his wife. He wanted to make the relationship work, to stay with her forever, however short their forever might be, but he couldn't raise another man's son. As a pureblood, even one so lax as he had been raised, the thought of a stranger's son inheriting his titles, of them passing out of the bloodline, was incomprehensible. The idea of seeing another man's face on the child who would call him 'father' was heart-breaking. Lily agreed that it would be unfair to her precious baby to make him live in a household where he would be resented by his parents, and she wanted him out of the war. Death Eaters had no qualms about attacking family.
Lily named her son Hayato, after an Order member who had flung himself in front of the Killing Curse to save her when she was pregnant. He had quietly befriended her, and helped her through the awful times when she would be trapped in the house by her pregnancy whilst her husband fought. The day he had died, she had insisted on leaving the house to shop. No attacks had been predicted. The Death Eaters had seemed to come from nowhere, and if not for Hayato, both mother and child would have perished.
With a heavy heart, she gave her son to Aunt Rose, who promised to give him to Stefano, who she knew would love him like he was a legitimate son. And so Gokudera Hayato grew up in a wealthy mafia family. Lily visited a few times a year, whenever she could make sure that she wasn't followed. Her relationship with James was strained by her insistence on holding on to a relationship he thought she should leave behind, and not even the birth of her second child eased the tensions. She told her son nothing of magic, of the war that was growing more desperate by the day. When Death Eaters attacked, throwing her car off the side of a cliff, Hayato blamed his father and left home. Meanwhile, Dumbledore searched desperately for a child who could fit the prophecy that was fast becoming their only hope.