I think everything had a logicality to it if you think about it. After Hashirama, it wouldn't have been a good choice for an Uchiha to take over as after Madara, it was really a matter of when they would rebel not if. Tobirama was already a good candidate for Hokage as he was the one already dealing with the politics of the village. Acting as Hashirama's second in command, he already had a grasp of what was expected of him so it was only natural to pass on the title to him.
Hiruzen was picked by Tobirama although this was at a time when they were in danger of being attacked. Considering he had no time to really consider a better alternative, based on his experience with his team, the only logical choice to make at the time would be to hand over the title to Hiruzen. Danzo, Kagami, and the Cho-something kid weren't part of his team but rather they were assimilated with with his team briefly. Considering they were at a time of war, it is only logical that their real sensei must have been killed in combat or close to it.
Hiruzen was quite lucky but as Hokage, in my own opinion, he didn't do really anything one would call dynamic or progressive, but he did do something that his predecessors and those from other villages weren't able to do: Maintain stability. Given his approach to problems, he really did a great job of keeping people together and making sure everybody was happy. From what I know, he did make a couple of mistakes on his part, he allowed Danzo to handle the darker affairs of being a Hokage and followed the advice of his friends, who from further speculation throughout the series, weren't very helpful. I think those three really hindered Hiruzen but at the same time I can see a sort of tactful approach to his descision. Danzo would have defected were he not handed the position to handle some of the affairs of the leaf. Infering from when he was younger, he did seem to possess around the same thought process as Hiruzen but as time grew he became colder and more distant. Being the leader of ROOT gave him the possibility of being a pseudo-hokage and having the eventual force to one day overtake the village.
The next choice after Hiruzen was Minato obviously. He had the talent/power as well as the mind of a Kage. He was easy to get along and judging from his casual attitude, he could have been easily manipulated. However, it came as a surprise to everyone that he actually was very intelligent (aside from naming things which he has a bad rep for xD). His approach to circumstances and his skills were a real deterrent for other nations not to mess with the leaf. Logically he would be a good choice both because of his skills and because he was close to the third so the transition would be a smooth one.
We already know about the other Hokage but one thing I can imagine they really considered was the transition process. In a world parallel to our own, "time is money". They wouldn't have picked someone who wasn't well acquainted with the Hokage role. It isn't just about fighting ability which the Daimyo care little about. Their foremost concerns would have been:
- How long is it going to take them to get assimilated?
- What do they know about being a Hokage (concerning paperwork)?
- Are they able to defend the interests of the Daimyo?
A large part of what we don't get to see is how they politics work in the Narutoverse but from we saw during the time that Tsunade spoke with them was that they function just like in our world, to a degree.
