I was watching the first GoT episode and I remember reading or hearing somewhere that the scene in the forest with the dead stag and direwolf is a foreshadowing device used by the writers. I know that Jon mentions that the 5 direwolf pups can represent the 5 stark children. But then Jon finds the albino direwolf. Considering the white fur, is this foreshadowing of Jon's true lineage? Is there any proof that this is what Martin or Weiss/Benioff intended?
Possibly, though only with broad strokes to signify Jon is different.
The dead direwolf and the pups forms part of a foreshadowing for the whole of book 1/season 1 in that Ned going down south will ultimately kill him. An antler killed the direwolf and we know a the direwolf is of House Stark and the stag of House Baratheon. It even worries Catelyn in universe when she thinks of the signs this sends.
It took Ned a moment to comprehend her words, but when the understanding came, the darkness left his eyes. "Robert is coming here?" When she nodded, a smile broke across his face.
Catelyn wished she could share his joy. But she had heard the talk in the yards; a direwolf dead in the snow, a broken antler in its throat. Dread coiled within her like a snake, but she forced herself to smile at this man she loved, this man who put no faith in signs. "I knew that would please you," she said. "We should send word to your brother on the Wall."
A Game of Thrones, Catelyn I
However, there is some significance to Ghost compared to the rest of the group, it is meant to show Jon as an outsider, the bastard Snow and not a Stark. Ghost is found away from the others having been shunned away and of course has white "snowy" fur.
"There," Jon said. He swung his horse around and galloped back across the bridge. They watched him dismount where the direwolf lay dead in the snow, watched him kneel. A moment later he was riding back to them, smiling.
"He must have crawled away from the others," Jon said.
"Or been driven away," their father said, looking at the sixth pup. His fur was white, where the rest of the litter was grey. His eyes were as red as the blood of the ragged man who had died that morning. Bran thought it curious that this pup alone would have opened his eyes while the others were still blind.
"An albino," Theon Greyjoy said with wry amusement. "This one will die even faster than the others."
Jon Snow gave his father's ward a long, chilling look. "I think not, Greyjoy," he said. "This one belongs to me."
A Game of Thrones, Bran I
Of course the Stark banner is of a grey direwolf, which all the main Stark's direwolves were, so this is again to show the Jon is different. I don't think it is in anyway to give a hint of who he really is just that he's not quite a Stark. Of course, as noted by Bran, there is some significance to Ghost having already opened his eyes but I'm not quite 100% sure what this could be signifying.
Just to note that there were 5 direwolf pups initially found 3 male and 2 female as there are 3 male and 2 female trueborn children of Ned and Catelyn and so there is significance to the pups themselves. Which, as previously mentioned, translates to Ghost being separated and different from the rest, as Jon is.
"Lord Stark," Jon said. It was strange to hear him call Father that, so formal. Bran looked at him with desperate hope. "There are five pups," he told Father. "Three male, two female."
"What of it, Jon?"
"You have five trueborn children," Jon said. "Three sons, two daughters. The direwolf is the sigil of your House. Your children were meant to have these pups, my lord."
Bran saw his father's face change, saw the other men exchange glances. He loved Jon with all his heart at that moment. Even at seven, Bran understood what his brother had done. The count had come right only because Jon had omitted himself. He had included the girls, included even Rickon, the baby, but not the bastard who bore the surname Snow, the name that custom decreed be given to all those in the north unlucky enough to be born with no name of their own.
A Game of Thrones, Bran I
It's also worth noting Ned's words earlier about Ghost "Or been driven away". It could be that the mother drove Ghost away from the others as she lay dying just as Cat had been driving Jon away for a long time because he is "different".
"What of Jon Snow, my lord?" Maester Luwin asked.
Catelyn tensed at the mention of the name. Ned felt the anger in her, and pulled away.
Many men fathered bastards. Catelyn had grown up with that knowledge. It came as no surprise to her, in the first year of her marriage, to learn that Ned had fathered a child on some girl chance met on campaign. He had a man's needs, after all, and they had spent that year apart, Ned off at war in the south while she remained safe in her father's castle at Riverrun. Her thoughts were more of Robb, the infant at her breast, than of the husband she scarcely knew. He was welcome to whatever solace he might find between battles. And if his seed quickened, she expected he would see to the child's needs.
He did more than that. The Starks were not like other men. Ned brought his bastard home with him, and called him "son" for all the north to see. When the wars were over at last, and Catelyn rode to Winterfell, Jon and his wet nurse had already taken up residence. 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. She might have overlooked a dozen bastards for Ned's sake, so long as they were out of sight. Jon was never out of sight, and as he grew, he looked more like Ned than any of the trueborn sons she bore him. Somehow that made it worse. "Jon must go," she said now.
A Game of Thrones, Catelyn II
Lastly, it is worth noting that George R. R. Martin employs a three step foreshadowing process and this is almost certainly the first step in that process that something is different with Jon. As such it is only signifying something is different with Jon through very general broad strokes rather than pointing to where he possibly came from.
Q: Anne, although you're the envy of many a GRRM fan, do you ever wish you didn't have to edit the books so that you could be surprised by them all at once along with the rest of us?
A: No. As above, he doesn’t tell me a lot. He feels I am most effective at my job if I am surprised along with everyone else. And it is easier to tell when he’s overplaying a hand and revealing things too early if you don’t actually know going in what will happen. That said, now that I’ve realized his three-fold revelation strategy, I see it in play almost every time. The first, subtle hint for the really astute readers, followed later by the more blatant hint for the less attentive, followed by just spelling it out for everyone else. It’s a brilliant strategy, and highly effective.
Quote taken from this answer and originally from universe.suvudu.com but I can't seem to find the interview on there anymore. However, I have seen it in the past and I've seen it quoted in various places before so I believe it is accurate.
There's very little I can add to LethalCarrot's superb answer. But there are a couple of points hinted or brushed past.
the pup was driven away: dying Lyanna gave away baby Jon to her brother
the pup opened his eyes: Jon throughout the series has kept his eyes on his own moral compass and vision of the world instead of following rules or obvious counsel. Tons of examples, ranging from not turning in Tarly when the latter ran off, having sex while a member of Night's Watch, insisting on going to meet Daenerys when the gathered council of the north told him he needed to stay, and finally trying to prevent the destruction of King's Landing.
regarding the three-fold pattern of revelation: I, too, only saw the sixth pup as a remark that Jon was actually a member of the family. The hint that he might be Ned's nephew was way, way too subtle a first step. The second step would be Bran's vision of the dying Lyanna giving her baby to Ned to raise as his own. The third step would be Tarly finding the annulment and new marriage in the archives.