In the original story, Severus often blames his Muggle father for the difficulties in his home life. But what if, instead, he internalized that blame? What if he believed that he was the reason for the rift between his parents, as many children do? This self-blame could change his behavior dramatically, particularly in his interactions with Lily and Petunia.
Imagine their first meeting with Severus feeling responsible for his family's dysfunction and desperately wanting to avoid causing any distance between the two sisters:
“H-how… how are you doing that?” Petunia asked, a mix of awe and jealousy in her voice.
Severus gently placed his hand under hers, and the bud in her palm blossomed into a beautiful petunia flower.
“Be my friend,” he said softly, his voice tinged with shyness. A tentative, awkward smile played on his lips, his attempt at confidence endearing in its uncertainty. “I’ll show you everything.”
EDIT: Severus would end up in Ravenclaw because he wants to make magical items so that Petunia can fun with him and Lily. This curiosity and desire to learn more about the process could get him sorted into that house.
He would regularly write letters to her, telling her all about the things they are learning so that she doesn't feel left out and can chip in when he and Lily are talking about spells and stuff.
Maybe even send her the occasional magical items he finds in hogsmead that function on their own and don't really need a wizard to work. Perhaps magic may be needed to recharge them, this may cause Lily and Severus to be treated as her item chargers.
