Was Jon Arryn capable as the Hand? What were his achievements?

Question

Jon Arryn is normally believed to have been a wise and capable administrator in the fandom. Generally there are two reasons given for that assumption:

1. He kept the peace for most of his tenure.
2. He gave the realm years of plenty.

But it appears to me that keeping the peace wasn't really much of an achievement. Westeros had just recovered from a horrific civil war and evidently everyone was too exhausted to start another war on the continent. Off-Continent, the Greyjoys, who mostly stayed out of Robert's Rebellion, except for the minor raids off the Tyrell coast, did rebel and Jon Arryn isn't mentioned at all in the campaign and it appears that King Robert, Eddard Stark, Stannis Baratheon and Tywin Lannister sorted out the Greyjoy rebellion without any help from Jon Arryn. His only real contribution seems to be keeping Dorne at peace after Robert's accession but then again, we can say that it was only because Doran plays to win, not because Arryn gave him any hopes for justice. Securing Cersei Lannister's hand for Robert can be considered an achievement as this marital alliance was the steel that kept the Seven Kingdoms together but then again, it's not like securing the proposal took a lot of effort, we all know Tywin always wanted Cersei to become the Queen.

As for the plentiful years, It seems like Petyr Baelish is to be praised/blamed for that. Petyr Baelish emptied the treasury that Aerys had left overflowing, although he blames the Hand and the King for that. Of course, Petyr's own aptitude for finance kept the economy floating on the surface even though Crown was drowning in domestic and foreign debts.

So the question is, is there any definite proof that Jon was a capable Hand? If yes, what were his achievements?

Answer 1

We as outsiders might not have any proof, but those who wrote the history think Jon Arryn performed his duties as Hand of the King quite remarkably.

First, prior to even being named Hand Jon Arryn was instrumental in starting and winning Robert's Rebellion. Then the Maester felt Robert's first great act was naming Lord Arryn his Hand.

In more recent years, the importance of the role played by Lord Jon Arryn in Robert's Rebellion cannot be gainsaid. Indeed, it was Lord Jon's refusal to deliver the heads of his wards, Eddard Stark and Robert Baratheon, that began the revolt. Had he done as he was commanded, the Mad King might yet sit the Iron Throne. Despite his advanced years, Lord Arryn fought valiantly beside Robert on the Trident. After the war, the new king proved his wisdom when he made Lord Jon Arryn his first Hand. His lordship's sagacity has helped King Robert rule the Seven Kingdoms wisely and justly ever since. It is a joy to the realm when a great man serves as Hand to a great king, for peace and plenty will surely come of it.

_{The World of Ice and Fire - The Vale: House Arryn}

The marriage pact he made with the Lannisters might not have taken much effort, but it was still a big deal as it sealed the peace that would follow;

As his first act, the unwed king took to wife the most beautiful woman in the realm, Cersei of House Lannister—thereby granting to House Lannister all the honors that Aerys had denied it. And though all know Lord Tywin might well have become Hand again, the king, in his graciousness, gave that office to his old friend and protector, Lord Jon Arryn, instead. The wise and just Lord Arryn has indeed helped the king shepherd the realm to prosperity since.

_{The World of Ice and Fire - The Glorious Reign}

Finally, since Lord Arryn's death the realm had devolved into war again clearly proving the Hand of the King had indeed done well when he lived;

The death of the noble Hand, Jon Arryn, has unleashed a madness on the land, a madness of pride and violence. The madness has robbed the realm of Robert, and of his fair son and heir Joffrey. Pretenders strive to steal the Iron Throne, and disturbing rumors of dragons reborn trickle in from the east.

_{The World of Ice and Fire - Afterword}

Now I would say all of this must be taken with a huge grain of salt as The World of Ice and Fire was written in-universe for King Robert himself.

However, "History is written by the victors!" therefore, Jon Arryn was indeed a competent Hand and anyone to say otherwise is merely speculating...

Answer 2

In addition to @Skooba's answer, Jon Arryn is also credited for setting the peace terms with Doran Martell after the rebellion was finished.

The Martells took a big hit with the brutal murder of Elia Martell and her children, and with the death in combat of Doran's uncle, Lewyn Martell, a sworn brother of Aerys' Kingsguard. Jon Arryn traveled to Sunspear (returning Lewyn's bones in the process) in a tentative to sign the peace terms with Doran.

Later in the books, we find out that this was all a long-term plan by Doran, where his acceptance of Jon's peace terms was just a means to an end, where he would later get revenge on the death of his daughter, grandchildren, and uncle.

Nonetheless, Jon Arryn is credited to get the Martells to bend the knee to Robert, and in a way or another, responsible to placate the Martell's fury.