Why was Snape the Half-Blood Prince?

Answer 1

It is a nick name he privately gave himself. It was due to the fact he was ashamed of the muggle half of his heritage. His mother was a witch and her maiden name was Prince. The name was his way of clinging on to her half of his heritage or blood-line.

Answer 2

"Half-Blood Prince" is a nickname Snape made up for himself when he was a student at Hogwarts. Snape's mother was a pure-blood witch named Eileen Prince while his father was a Muggle named Tobias Snape. Thus, Snape is literally the half-blood member of the Prince family. The Half-Blood Prince's book is Snape's Potions textbook from back when he was a student at Hogwarts.

Answer 3

He hated his muggle father and so decided to use his mothers maiden name, Prince and he is a half-blood.