Who was/is your favourite Hand of the King? Which do you think was/is the best in this role?
HJ, you've got two questions here and my answers are not the same.
Since the poll question is in regards to my favorite, I'll tell you... it's Tyrion. First, Tyrion is not really the Hand. He's Tywin's proxy. He's the Hand for all intents and purposes because Joffrey was weak and Cersei is not smart. He knew he'd be replaced sooner or later by his father... which is exactly what happened. Second, Tyrion has his own POV. Sure Eddard and Davos have theirs also, but I feel that so much more fascinating things went on during Tyrion's tenure. Third, Tyrion is simply my favorite character in this poll.
Your second question is in regards, as the title of the posts on this thread indicate, to the most competent Hand. This is a much more complicated issue. Different problems plagued the realm during the terms of these various Hands... and different qualities of leadership were required to solve them.
The POVs are used to affect our judgment of characters. Eddard and Jon were fairly selfless. Both seemed to have the vested interests of the kingdom in high priority. Davos also seems to desire justice. Like Jon and Eddard, he has an extreme amount of loyalty to his king. In contrast, Tywin is repeatedly called avaricious... by his own son! Yes, Tywin was competent in regards to the treasury, the military, domestic policy, and foreign relations, but it seems that he was also looking for opportunities to entrench his family at court. And after Aerys turned on Tywin, Tywin betrayed him to Robert.
Is this the truth or just a matter of influence from the POV? What was Tywin's big crime against Aerys? Oh yeah, he wanted to marry his daughter to the Crown Prince. But that's exactly what Eddard and Robert schemed to do! Tywin was fired, but Eddard got promoted!
Also, Eddard had a dislike for the economy and it seems Jon put the realm into ruin.
I think Tywin was the most competent Hand of the King.
Also, Davo's predecessor, Lord Florent was not mentioned in the list of Hands.
On a side note, I think it's interesting that neither Renly, Robb, Balon, Euron, nor Dany has nominated a Hand. As for the Ironborn, mayhaps it's not in their tradition. Mayhaps Dany has not really fleshed out her court nor decided that her kingdom is yet big enough to require a Hand. Robb was at war long enough to have discussed a Hand... mayhaps the Starks never had one in the past and mayhaps it was a bone to be offered as a bargaining chip for an alliance with Dorne, the Vale, or Highgarden. Renly quickly put together a large court complete with Kingsguard, councillors, and an army, but the glaring exception is a Hand. Now I could see Robb, Balon, Euron, and Dany wanting to personally rule, but Renly, imo, should not have missed the chance to have someone else "count coppers."