Is that the exact quote, because I could've swore it was more like... You might not have the Stark name, but you have Stark blood in your veins...
In the book, the quote isn't made to Snow but to Cat.
I have the book as a pdf on my phone, here's the relevant passage.
“That cut deep. Ned would not speak of the mother, not so much as a word, but a castle has no secrets, and Catelyn heard her maids repeating tales they heard from the lips of her husband’s soldiers. They whispered of Ser Arthur Dayne, the Sword of the Morning, deadliest of the seven knights of Aerys’s Kingsguard, and of how their young lord had slain him in single combat. And they told how afterward Ned had carried Ser Arthur’s sword back to the beautiful young sister who awaited him in a castle called Starfall on the shores of the Summer Sea. The Lady Ashara Dayne, tall and fair, with haunting violet eyes. It had taken her a fortnight to marshal her courage, but finally, in bed one night, Catelyn had asked her husband the truth of it, asked him to his face.
That was the only time in all their years that Ned had ever frightened her. “Never ask me about Jon,” he said, cold as ice. “He is my blood, and that is all you need to know. And now I will learn where you heard that name, my lady.” She had pledged to obey; she told him; and from that day on, the whispering had stopped, and Ashara Dayne’s name was never heard in Winterfell again.
Whoever Jon’s mother had been, Ned must have loved her fiercely, for nothing Catelyn said would persuade him to send the boy away. It was the one thing she could never forgive him. She had come to love her husband with all her heart, but she had never found it in her to love Jon.”
Excerpt From: George R. R. Martin. “The A Song of Ice and Fire Series.” Bantam Books, 2012-09-11. iBooks.
The quote from the series is Ned to Snow.
"
There’s great honor in serving the Night’s Watch. The Starks have manned the Wall for thousands of years, and you are a Stark. You may not have my name, but you have my blood.” 
