Why was Snape a Half-Blood Prince?

From the time Harry got his hand on the half-blood prince’s book, there were speculations among the trio(Harry, Ron and Hermione) whom the book belonged to.

While Harry pointed out that it must be some bloke, Hermione thought that it might belong to some witch.

She was right all along, the book belonged to Snape’s mother.

But, Snape told Harry his identity of Half-Blood Prince making no more clear why he chose such title for himself.

It was Hermione who found out the exact reason:

‘No – no – Harry, I didn’t mean that!’ she said hastily, looking around to check that they were not being overheard. ‘It’s just that I was right about Eileen Prince once owning the book. You see … she was Snape’s mother!’

‘I thought she wasn’t much of a looker,’ said Ron.

Hermione ignored him. ‘I was going through the rest of the old Prophets and there was a tiny announcement about Eileen Prince marrying a man called Tobias Snape, and then later an announcement saying that she’d given birth to a -‘

‘- murderer,’ spat Harry.

So, Snape is Half-Blood Prince as:

Mother – Witch Father – Muggle, thus earning him the half-blood title.

Eileen Prince being Snape’s mother and Prince being her surname. Snape went with name ‘Prince’ considering abusive nature of his father.

He had chosen the title in his teenage years. He was invisible, nobody seemed to like him. James Potter and Sirius used to mock him. So, he chose an identity which he seemed more connected with, which could justify for what he truly was and not how the world perceived him.

And Half-Blood Prince was born!