In the epic poem, Beowulf, what attributes made a good king in Beowulf's time?
From reading Beowulf, I think that the attributes that made a good king in Beowulf's time are almost as mythical today as is our hero, Beowulf.
In comparing the two kings in the story, we must study Hrothgar and Beowulf. Hrothgar is the leader of the Danes. He is admired for his bravery and his honor. He values the lives of his people, and will do whatever he can to protect them. He admires other men of valor and is appreciative of the help Beowulf offers:
...he adopts Beowulf "in his heart" as a son. [A great] tribute...[coming] from the mouth of a king.
Hrothgar is wise, a man of God, and one not afraid to show his emotions in public which is not seen as a weakness of that time, but simply one side of Hrothgar's nobility.
Beowulf shows the reader early on that he is a man of valor and honor. He is not as interested in what he may gain personally from his actions as he is in finding how he might be of service to others. In fighting Grendel, his desire is to aid the Danes, and bring glory to his feudal lord. He has concern for the lives of others and is willing to sacrifice his health and life to serve the needs of those in Hrothgar's kindgom. Beowulf is a man of God; his faith is strong and he recognizes that his successes depend on the will of God.
These characteristics follow Beowulf through the fifty year until he also becomes King of the Geats.
Beowulf's valor, hereditary pride and faith are not restricted to the days of his youth.
Even though he is now much older, he still exhibits the same valor of his younger days, and welcomes the opportunity to defend his people from the wrath of the "firedrake," even if he loses his own life. He also is a king who cares deeply for his people and is willing to sacrifice himself for the welfare of others. In death, he is still considered a "gracious and fair-minded" king.
In summary, a good king in the time of the story of Beowulf was a man who was honorable, loyal, wise, fair, someone of strong Christian faith, and someone concerned for his people—willing to do whatever was necessary for the good of others. His own welfare was not as important as his need to be of service to those in need: he was one willing to make sacrifices.
The other answer to this question does a great job of highlighting honor, faith, wisdom, and loyalty as important kingly qualities in the world of Beowulf. I'd like to add that kings were also expected to display generosity to their subjects by giving gifts. Hrothgar periodically provides his subjects with generous feasts, and he lavishes Beowulf with gifts on several occasions. After Beowulf defeats Grendel, for instance, Hrothgar holds a feast and gives rich accessories, including a sword, armor, and horses.
The giving of gifts was an important aspect of the world in which Beowulf takes place, ensuring the loyalty of a king's subjects and their continued service to the kingdom. In other words, this generosity was a key component of maintaining social stability, of keeping the community of Heorot intact and strong against the forces of chaos, represented by monsters like Grendel, lurking on the fringes of civilization. A good king was one who could be generous in rewarding the loyalty of those who served him, as this process ensured that a civilized community like Heorot could continue to exist.