Why did Jon Snow choose someone he barely knew to this post?

While there is some doubt about whether Davos is formally his Hand or even 'right hand man', or whether the King in the North even has 'Hand' as a title - it's still the case that he's adopted him as a close trusted advisor.

Hand of the King isn't specifically a military role. The role is probably closest to the historical role of a prime-minister in an absolute monarchy. The Hand exercises a lot of the monarch's power in governing their Kingdom, and at times of war probably help advise their monarch, but they are not necessarily Generals as well.

I think Ser Davos is quite modest about his military prowess. He knows how to fight, and does so alongside Jon from their time together at The Wall. He clearly knows a fair amount about sea warfare. He was the first to notice the danger of the wildfire defense of King's Landing (in the books anyway). He was clever enough to bypass the siege of Storm's End when he brought onions and food to the castle. Jon, who appears to have received the same military training as the other sons of Ned Stark (and Theon), is also comfortable with battle on land. To have a Hand who is comfortable at sea complements his skills well.

It is also worth noting that Ser Davos was also the chosen Hand of the King of Stannis Baratheon, who does not appear to be easily swayed by sentiment. I think Jon probably values the same attributes as Stannis Baratheon did. He is honest and plain speaking even with Kings - so his advice is likely to be honest and not just flattering them.

Other "Hands" seem to come from varying backgrounds. Tywin Lannister seems to be rewarded with the role as a key ally to the Targaryen as recognition on his wealth and power that supports the crown. Second time around it is in recognition that Joffrey is half-Lannister and that the Lannister's rescued King's Landing from the rebellion. Ned Stark is loyal to Robert Baratheon, and a strong military man, but not necessarily a proven administrator or politician. Tyrion is shrewd and well educated, a problem solver and a good reader of people and Daenerys already has strong military advisors.

This question seems to erroneously imply that Davos is the Hand of the King, which does not appear to be the case. The fact that Jon heeds his council does not make him Hand of the King. The Hand of the King reigns in the King's absence, which would make Sansa Stark a far better fit for that role.

I can only assume that you think he's the Hand of the King because Jon specifically asked Ser Davos to come along to Storm's End, but there's a far better reason for that: Ser Davos knows everything there is to know about Storm's End. He's a smuggler, if things go sour, he's Jon's best bet at getting out of the castle alive. Note that they are sailing to Storm's End, and Davos has managed to bypass a blockade before. Taking him along on the trip is not a sign of Davos being Hand of the King, it's a sign of Davos being a good sailor and smuggler.