Who killed Jon Arryn and why?
Littlefinger conspired with Lysa to kill Jon Arryn (the lord of the Vale) by giving Lysa poison, which she used to murder Jon.
Littlefinger had Lysa tell Ned and Catelyn Stark that the Lannisters poisoned Jon, leading to further animosity between the houses that later erupted into war.
Lysa killed Jon Arryn with poison at Littlefinger’s behest.
Lysa’s motive was that she was absolutely fanatical about protecting her son, to the point where she refused to be parted with him.
You can see this overprotectiveness when Catelyn brings Tyrion to the Eyrie.
Jon had wanted to foster Robert Arryn on Dragonstone with Stannis Baratheon (or at Casterly Rock with Tywin, depending on whose story you believe), a common practice for highborn boys to make social contacts and learn about combat and arms training.
Lysa had no intention of letting Robert be sent away, which gave her the motivation to kill Jon.
Who killed Jon Arryn and why?
Littlefinger’s motive is more complex. He knows that Robert Baratheon trusts Jon. He also knows that Jon is close on the trail of Jaime and Cersei and is nearly ready to tell Robert about their affair.
If the Lannisters are pushed out of power, that just solidifies the Baratheon-Arryn-Stark-Tully power bloc, in which case there’s less opportunity for someone like Baelish to rise.
Baelish sows the mistrust and chaos and then takes advantage of the opportunity that chaos creates. Baelish is also still interested in “payback,” of sorts, for being denied Catelyn’s hand in marriage and for always being reminded of his low birth and status.
So his path sees him double-cross Ned, who got to marry Catelyn, end up married to Lysa and in power in the Vale, his home region, and become lord paramount of the Riverlands, spite to House Tully, the family that fostered him and the family that shot down his marriage ambitions.
Killing Jon has a twofold effect:
It helps him begin to “payback” the Starks, Tullys, and Arryns for his perceived insults, and it allows him to rise in power at their expense.
I don’t believe that Varys actually killed Jon, but he knew who did it and didn’t speak up. He gives hints to Ned about the killer’s identity: the boy who owed all he had to Jon Arryn.
This is when they’re generally talking (and in Ned’s case, thinking) about Ser Hugh, who has just died; Ned thinks Vary is talking about Hugh when really he’s very broadly pointing to Littlefinger, who was also once a boy who owed his social rise to Jon’s patronage.
I think this is what leads some people to think that Varys is claiming responsibility in his conversation with Illyrio when really, all he did was allow Littlefinger and Lysa to go undetected.
The same sort of thing holds true for Pycelle: He didn’t kill Jon himself, but nor did he actively try to keep him from dying.
Who killed Jon Arryn and why?
The Arryn murder is an interesting case, where we’re actually told more or less who did it very early on: Poison is a “woman’s weapon,” and Lysa and Cersei were really the only two women with such broad access to Jon.
You can discern from what Bran overhears that Jaime and Cersei didn’t kill Jon; they’re just concerned that Lysa knows about their affair and will try to pin the murder on them or publicize their affair or both.
Lysa has said that she’d kill anyone who tried to take her son from her, which, in her mind, included Jon when he wanted Robert to be fostered.
Combine that with Varys’ talk about the young boy who owed Jon everything he had, and there are your murderers.
Everything else — the Lannisters being implicated, Ser Hugh’s convenient death — was a red herring or a coincidence.
In conclusion, it seems that Varys and Lysa have both taken credit for Jon’s death, and neither of their cases can easily be explained away. At this point, it seems a little more likely that it was Lysa who killed Jon, and Varys is simply lying or stretching the truth.
Here’s the logic for Varys:
In A Game of Thrones, pg. 434, Arya overhears a conversation between Illyrio Mopatis and Varys.
“If one Hand can die, why not a second?” replied the man with the accent and the forked yellow beard. “You have danced the dance before, my friend.”
…”Before is not now, and this Hand is not the other,” the scarred man said as they stepped out into the hall.
This seems to heavily imply that Varys played a role in Jon Arryn’s death. He certainly had a motive–Varys and Illyrio are plotting to bring the war to Westeros so as to weaken the land for a Targaryen invasion, and Jon’s death allowed Eddard to come to King’s Landing and would eventually start the war.
(For a full discussion of Varys’s motives, see Who were the two men Arya saw conspiring underneath the Red Keep? What are they plotting?) Also, Jon Arryn was poisoned by the tears of Lys… and Varys happens to be born in Lys.
All this would make sense except for the fact that Lysa Arryn also admits to the murder (see below).
This leaves us with several options. One is that the murdered Hand Illyrio is referring to is not Jon Arryn, but another.
The problem is that no other Hand fits the profile. Of the Hands that ruled while Varys was in King’s Landing, Lord Merryweather and Lord Connington were exiled, Lord Chelsted was executed by Aerys after refusing to burn down the city with wildfire, andRossart was killed by Jaime in the Sack of King’s Landing.
Another possibility is that Varys subtly maneuvered Lysa into doing the deed. On AGOT p344, he says that Lysa Arryn has fled beyond his reach, implying that he once had her in his grasp. Perhaps Littlefinger did not tell Lysa to kill Jon at all; instead, it was a clever manipulation by Varys.
The third, most likely possibility is that Varys did not kill Jon but is taking credit for the deed all the same.
Thus, Varys seems a likely suspect, but Lysa Arryn’s confession (see below) rather confuses the issue.
Here’s the logic for Lysa:
Near the end of A Storm of Swords, on page 913, Lysa goes into hysterics and starts yelling at Littlefinger:
“Tears, tears, tears,” she sobbed hysterically. “No need for tears… but that’s not what you said in King’s Landing. You told me to put the tears in Jon’s wine, and I did. For Robert, and for us! And I wrote Catelyn and told her the Lannisters had killed my lord husband, just as you said.”
If Lysa did poison Jon at Littlefinger’s behest, it would explain a lot of things.
Littlefinger’s motives are unclear, but it seems that he has been plotting war, and the death of Jon Arryn fits in with other events (like his blaming Bran’s attempted assassination on Tyrion) that he has had a hand in.
Also, Lysa says that the tears of Lys were used, and it seems very unlikely that she would know this if she had not done the deed.
All this is well and good, but what about Varys’s confession to the same crime (see above)?
Lysa has no reason to lie about her role in killing Jon; on the contrary, she shouldn’t even be taking credit for it with Sansa and Marillion as witnesses. Varys, on the other hand, may very well have been lying.
Also, note that Littlefinger never confirms to Lysa that it was him who told her to murder Jon–it might very well have been a manipulation of Varys.
However, this seems to be pushing things a little too far. A plot twist in which Littlefinger convinced Lysa to kill Jon is believable.
A plot twist in which Varys pretends to be Littlefinger convincing Lysa to kill Jon is a bit of a stretch.