Books' Perspective
Who was responsible for commencement Jon Arryn's investigation?
It was Lord Stannis who suspected something wrong in physical features of his nephews and niece first and brought his suspicions to Jon Arryn, which lead to their investigation about parentage of the three princelings.
As Stannis explained to Catelyn in A Clash of Kings:
"Lord Stannis," she asked, "if you knew the queen to be guilty of such
monstrous crimes, why did you keep silent? "
"I did not keep silent," Stannis declared. "I brought my suspicions to
Jon Arryn"
Then he explains why he came to Jon Arryn with his suspicions rather than Robert himself:
“Rather than your own brother?”
“My brother’s regard for me was never more than dutiful,” said
Stannis. “From me, such accusations would have seemed peevish and
selfserving, a means of placing myself first in the line of
succession. I believed Robert would be more disposed to listen if
the charges came from Lord Arryn, whom he loved.”
Jon Arryn had no idea or suspicions prior to Lord Stannis coming to him with his suspicion. Or at least, nothing suggests that he in fact had any suspicions before Stannis approached him.
The investigation
Jon Arryn and Stannis Baratheon worked closely to investigate the apparent lack of Baratheon features in Cersei's children. And at first Jon began by obtaining the Lineage of Great Houses from Grand Maester and observing animal breeding to understand how genetic traits were passed from parents to off-spring.
"Carrots and apples," Ned repeated. It sounded as if this boy would be even less use than the others. And he was the last of the four Littlefinger had turned up. Jory had spoken to each of them in turn. Ser Hugh had been brusque and uninformative, and arrogant as only a new-made knight can be. If the Hand wished to talk to him, he should be pleased to receive him, but he would not be questioned by a mere captain of guards … even if said captain was ten years older and a hundred times the swordsman. The serving girl had at least been pleasant. She said Lord Jon had been reading more than was good for him, that he was troubled and melancholy over his young son's frailty, and gruff with his lady wife. The potboy, now cordwainer, had never exchanged so much as a word with Lord Jon, but he was full of oddments of kitchen gossip: the lord had been quarreling with the king, the lord only picked at his food, the lord was sending his boy to be fostered on Dragonstone, the lord had taken a great interest in the breeding of hunting hounds, the lord had visited a master armorer to commission a new suit of plate, wrought all in pale silver with a blue jasper falcon and a mother-of-pearl moon on the breast. The king's own brother had gone with him to help choose the design, the potboy said. No, not Lord Renly, the other one, Lord Stannis.
A Game of Thrones - Eddard VI
Evidently whatever they found during their reading and observations was suspicious enough so they started seeking out natural born Children of Robert Baratheon. The information however didn't mean much for Eddard since he didn't understand much of it particularly the visit to Tobho Mott to see Gendry on pretext of commissioning a new armour.
From A Game of Thrones:
“The lad swears Lord Jon was as strong as a man half his age. Often
went riding with Lord Stannis, he says.”
Stannis again, Ned thought. He found that curious. Jon Arryn and he had been cordial, but never friendly. And while Robert had been
riding north to Winterfell, Stannis had removed himself to
Dragonstone, the Targaryen island fastness he had conquered in his
brother’s name. He had given no word as to when he might return.
“Where did they go on these rides?” Ned asked.
“The boy says that they visited a brothel.”
“A brothel?” Ned said. “The Lord of the Eyrie and Hand of the King
visited a brothel with Stannis Baratheon?
They had gone to that brothel to see Barra, Robert's daughter. Despite the mother having light colored hair, the daughter had inherited Baratheon black locks which confirmed some of their suspicions.
Then Ned went to the armourer Jon Arryn and Stannis Baratheon went to together.
Ned smiled. “Did you make a falcon helm for Lord Arryn?”
Tobho Mott paused a long moment and set aside his wine. “The Hand
did call upon me, with Lord Stannis, the king’s brother. I regret to
say, they did not honor me with their patronage.”
Ned looked at the man evenly, saying nothing, waiting. He had found
over the years that silence sometimes yielded more than questions. And
so it was this time.
“They asked to see the boy,” the armorer said, “so I took them
back to the forge.”
There they found out Gendry, whose mother was a blonde yet the boy had Baratheon Black hair and blue eyes. This further cemented their suspicions.
Not to mention, Stannis knew that Robert's bastard Edric Storm who was sired on Delena Florent also had black hair and blue eyes. He sent the child to Storm's End after the birth. Jon Arryn plausibly knew1 about the bastard daughter Robert had fathered when he was a ward of Lord Arryn at Eyrie. That Girl, Mya Stone also had black hair and blue eyes.
The final thing that confirmed their doubts was what they found out in a book about lineage of Great Houses which recorded marriages and births.
“One thing,” Ned told him. “I should be curious to examine the book
that you lent Jon the day before he fell ill.”
“I fear you would find it of little interest,” Pycelle said. “It was
a ponderous tome by Grand Maester Malleon on the lineages of the great
houses.”
“Still, I should like to see it.”
There, Lord Arryn found out that every time a Baratheon married a Lannister, Black hair genes won against Golden hair genes. This was the final proof they needed. As Eddard Stark later found out for himself:
The seed is strong, Jon Arryn had cried on his deathbed, and so it
was. All those bastards, all with hair as black as night. Grand
Maester Malleon recorded the last mating between stag and lion, some
ninety years ago, when Tya Lannister wed Gowen Baratheon, third
son of the reigning lord. Their only issue, an unnamed boy described
in Malleon’s tome as a large and lusty lad born with a full head of
black hair, died in infancy.
Thirty years before that a male Lannister had taken a Baratheon
maid to wife. She had given him three daughters and a son, each
black- haired.
No matter how far back Ned searched in the brittle yellowed pages,
always he found the gold yielding before the coal.
Littlefinger's role
In the same Book, Varys also said that Stannis had read the book and examined the physical features of Robert's bastards, although he mentions it in the wrong order as Stannis' own statements suggest that he was in fact the first person to suspect that, rather than the last.
"He accuses my brother and sister of incest. I wonder how he came by
that suspicion."
"Perhaps he read a book and looked at the color of a bastard's
hair, as Ned Stark did, and Jon Arryn before him. Or perhaps someone
whispered it in his ear." The eunuch's laugh was not his usual giggle,
but deeper and more throaty.
"Someone like you, perchance?"
"Am I suspected? It was not me."
Tyrion then asked Varys if it was not him, then who lead Stannis to his investigations and sowed the seeds of the suspicion. Varys subtly hinted that it may have been Littlefinger.
"If you were not this whisperer, who was?"
"Some traitor, doubtless." Varys tightened the cinch.
"Littlefinger?"
"I named no name."
Notice how Varys does not deny that it was Littlefinger but instead just insists that he named no one. Of course, Varys has his own motives so everything he says can't be trusted.
So, it is possible that it was Littlefinger who made Lord Stannis suspicious about parentage of Cersei's children and therefore planned the war of the five Kings.
Will Stannis seek something to claim the throne for himself regardless?
That would be against the very character of the man. He had no desire to claim the throne. But once he came to find out that he was actually the rightful heir, he was determined to win it because:
- He has an iron sense about laws. If something is his by law, he will never forfeit it. If something wasn't his by law, he would never claim it.
- He had a duty to his daughter who was the rightful heir to the throne after him.
- He believed he had to fight the war for survival of humanity even though he thought that this war will eventually kill him. But he was willing to pay that price for his people.
As you see, the desire for the throne or greed rather, doesn't come into it.
To prove these points, see the following canon evidence.
Stannis said this to Davos in A Storm of Swords, when the latter asked him why did he want the throne:
It is not a question of wanting. The throne is mine, as Robert's heir. That is law. After me, it must pass to my daughter,
unless Selyse should finally give me a son. I am king. Wants do not
enter into it. I have a duty to my daughter. To the realm. Even to
Robert. He loved me but little, I know, yet he was my brother. The
Lannister woman gave him horns and made a motley fool of him. She may
have murdered him as well, as she murdered Jon Arryn and Ned Stark.
For such crimes there must be justice. Starting with Cersei and her
abominations. But only starting. I mean to scour that court clean. As
Robert should have done after the Trident.
He also explained that he was willing to pay the ultimate price as was his duty being King:
I know the cost! Last night, gazing into that hearth, I saw things in
the flames as well. I saw a king, a crown of fire on his brows,
burning… burning, Davos. His own crown consumed his flesh & turned
him into ash. Do you think I need Melisandre to tell me what that
means? Or you?
If Cersei had given birth to even one Black-haired child2, Stannis probably would have never doubted anything. He was not looking for excuses to claim the throne.
Conclusion
One thing is clear. It was Lord Stannis who started the investigations after taking Jon Arryn on board.
Another thing, which is unclear but is hinted, is that Littlefinger may have caused Lord Stannis to be suspicious in the first place.
Show's Perspective
In the show, like many other key logical variables, this one is also missing.
It is implied that Stannis had no idea about the truth behind birth of Cersei's children and only learnt about it when Eddard Stark sent him a letter informing him about death of Robert and him being the heir.
From Show Wiki:
Stannis hosts a council and prepares a letter to be distributed
throughout the Seven Kingdoms. He has learned from Eddard Stark that
Joffrey Baratheon is a bastard born of incest between Cersei
Lannister and her brother Jaime rather than Robert Baratheon's true
heir.
Show does not give any hint that why did Jon Arryn begin his investigations except that he did and died as a consequence. There is no mention of liaison between Stannis and Jon Arryn in the show.
1. Since whole Vale knew who Mya's father was, How could Lord of Vale not know that it was Robert, who was his own ward?
2. In Books, Cersei never gave birth to any Black haired child or any child whose father may have been Robert. Show however, added a brief mention of a Child who died in infancy who may have been Robert's. Which begs the question that if that was the case, Why should have Jon Arryn suspected anything wrong at all? The suspicious thing was that none of the Princelings had Black hair. If there was one, it should have quelled the suspicions as suggested by Tyrion in the books.
