The Half-Blood Prince is Severus Snape. He called himself this because his mothers maiden name was Prince, but his father's surname was Snape, so he is half of the Prince's bloodline. There you have it: Half Blood Prince. His father was also a muggle making the Half-Blood part work as well.
Tobias Snape - father, muggle Irma Prince - mother, witch
Severus Snape - son, disliked his abusive father, and was proud of his wizarding heritage through his mother. He's a half-blood, and at some point prefered to be considered a Prince, rather than a Snape.