Intelligence
Every Hokage needs to be pretty smart, able to identify problems and come up with effective solutions.
Shino's pretty smart, pretty clearly. He identifies his opponent's weak points pretty much immediately
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and
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, and exploits them just as quickly. He's also got great improvisation skills and instincts,
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Yes, instincts are a feat.
[*]Knowledge/Wisdom
Knowledge just means things like creating/knowing/understanding a variety of jutsus and things about how the ninja world works. This would include things like inter-village relations, the history of the village's clans, etc.
Wisdom refers to understanding the ideals and beliefs that have been passed down from Hokage to Hokage. Every Hokage has come to understand the importance of training and protecting the next generation, so that the next generation can uphold the village's accomplishments and ultimately succeed the elders. Every Hokage also knows not to underestimate opponents/young shinobi that appear to be struggling. Etc.
Shino's also very knowledgeable.
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Also, his feat against Zaku required a good understanding of how chakra, sound, and air pressure work. Pretty impressive for a style of jutsu that was new to Konoha. Furthermore, he's someone who's probably going to advance his own clan jutsu:
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Shino was possibly the wisest member of the K12 early on, and he was able to articulate his beliefs more eloquently than any member of the K12.
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. Humility is important in a Hokage; it means they'll be willing to listen to advice as well. Also, we can assume that his role as a teacher means he intends to and is capable of imparting this wisdom to the next generation of Shinobi. Speaking of which, the fact that he becomes a teacher implicitly shows that he's internalized some of the previous Hokage's precepts, choosing to train the next generation directly.
[*]Diplomacy
This is pretty obvious; the Hokage should be able to negotiate between clans and between villages pretty well. Again, every Hokage has been pretty good at this.
For Shino, diplomacy is a given. He's a member of the Aburame clan, meaning he probably witnesses interclan politics, or at least understands their importance. He also knows how to keep calm in any given situation. Plus, as stated previously, he's pretty eloquent when he does decide to talk. Above all else, he knows when he doesn't need to talk. Dude doesn't have any real trigger points that would set him off. Aside from this, again, being a teacher should mean you gotta be good at solving conflicts.
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Shino wasn't even trying. Dude is a natural.

[*]Strength/Skill
This is pretty obvious; the Kage is usually the strongest ninja in the village.
Like I've said before, Shino was one of the strongest shinobi among the initial K12. I don't wanna get into things like matchups too much, but he has an answer to anything any of the K10 can do; his bug jutsu is incredibly versatile. At the very least, he's a very formidable shinobi.
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Skill-wise, like I've said, he's likely to advance his jutsu further. Furthermore, his jutsu's abilities would be amazing for an Hokage. The bugs can be used to provide intel to Shino. It's unlikely that any other Hokage would be better at avoiding being tricked or subverted the way Danzo subverted the Third and Fifth. This also helps with diplomacy. Nobody's going to really care if there's a stray insect in the room; after any given meeting, Shino can just tune back in with the bugs and learn anything he might have missed. Surveillance during important events, like the Chunin exams, would be no issue.
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It's just an incredibly useful jutsu for a Hokage to have.
Just a note, but
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Prestige
This is maybe a little controversial, but by "prestige" I just mean that he ninja should have built up a reputation within the village. They should be someone respected and trusted by most of the inhabitants.
As far as prestige goes, the dude's a member of the Aburame clan, one of Konoha's distinguished clans. He's also at least as well-respected as the other K12, but early on he was considered one of the most skilled shinobi among them. Undoubtedly he's known by most of the village, especially since he's teaching their children.
[*]Self-sacrifice
Pretty obvious: Every Hokage has died, or been willing to die, for his or her comrades.
Shino didn't even hesitate to try to stop Kankuro (of course, he was also spoiling for a fight, which I think is awesome) during the Konoha Crush. Even while poisoned, he fights on.
On a less drastic level, Shino's very brave and decisive - he immediately tried to take down Tobi as soon as Tobi confronted Kakashi's scouting team and Kakashi gave the okay.
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Again, my man was spoiling for the fight. Fears NOTHING.
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when Kakashi assigned Naruto, Sakura, Shikamaru, and one of the ninja dogs to find Sasuke.
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; he was just concerned about the mission (and getting his fight with Kankuro).