When Professor Severus Snape was trying to teach Harry occlumency in Harry Potter and the Order of the Phoenix Harry gets curious about the memories that Professor Snape had tried to hide from him, and he sees a scene where Severus Snape is bullied by Harry’s Dad when they were teenagers at Hogwarts.

Harry also sees his Mum who was described by Professor Remus Lupin as an uncommonly kind person try to stop the bullying, and Remus Lupin said to Harry that he regretted that he wasn’t a friend to Severus Snape.

As Remus Lupin told Harry that Severus could’ve tampered with The Wolfsbane Potion to stop Remus turning into a werewolf while he taught at Hogwarts, but Professor Snape didn’t. Remus Lupin also knew that Professor Albus Dumbledore chooses to trust people by giving them second chances when other people wouldn’t.

I also think Remus trusted the headmaster’s judgement on people’s characters as he himself felt that Albus Dumbledore gave him a chance.

Even though the headmaster knew that Remus was a werewolf.

I also feel that Severus Snape deeply regretted calling Harry’s Mother a mudblood as it wasn’t based on his true feelings and that he said “mudblood” in the heat of the moment.

As it would’ve been a very embarrassing moment for him to be in the air and upside down in front of not only the whole school but the girl he loves.

So, I think Severus Snape started calling himself the Half Blood Prince sometime after Lily had ended their friendship and telling Severus that she won’t bother defending him anymore. Then another reason why she ended their friendship is because she knew that Severus was going to be a death eater because at the time he was around people that he’d considered were his friends that she didn’t approve of which would’ve really hurt.