It is possible that Eileen had siblings or cousins who married and continued the family name. But if she did not and she was the last Prince, then the Prince family died out with her, and the female line of the family became extinct upon the death of her son Severus Snape, who never had children.
Prince is a general term for a monarch or a member of a monarch's family, and is a hereditary title of some European royalty, likely alluding to the aristocratic status of pure-blood families in the wizarding world. The word is derived from the Latin princeps, meaning "first", which was one of the titles of the Roman Emperor (as princeps civitatis, which means "first citizen"). The Prince is also the title of a famous political treatise by Niccolò Machiavelli; an interesting point considering that the adjective derived from the author's surname, Machiavellian, is defined as "attempting to achieve goals by cunning, scheming and unscrupulous methods", which may allude to the ideal traits of Slytherin house.
- In Harry Potter and the Half-Blood Prince, Chapter 30 (The White Tomb), Harry speaks of Snape playing up "pure-blood mother, Muggle father" in response to an article Hermione is reading when she says Snape's father was a muggle. However, the article never specifically said Eileen was pure-blood. There is a chance Harry made an assumption when hearing of Tobias. Snape would still be half-blood if his mother was as well.
- Wiktionary Definition: Machiavellian