Answer:
Zeus, who swung both in the sky as a reward for saving the god's life.
(But there are many other versions)
Explanation:
Actually, there are way too many myths, even among old Greek myths, that could explain the origin of the two constellations.
The constellation Ursa Minor, when including stars less visible, but still observable with the naked eye, vaguely resembles a bear (with an unusually long cause). Thus, in conjunction with the neighboring Ursa Major, it forms the basis of the myth of Callisto. It is said that the tail has been prolonged, since the bears do not have tails so long, due to the incessant rotation of the bear by the tail, around the pole.
The variant of the story, in which the constellation of Boieiro corresponds to the character of Arcas, tells that Ursa Minor represented a dog. This is an older tradition that sensibly explains both the tail length and the already obsolete alternative name of Cynosura (the dog's tail) to Polar. (It also clarifies the otherwise inexplicable etymology of the word Cinosura)
In even more remote times, the Ursa Minor had only seven stars, and were mythologically seen, and thus, as sisters. It is thought that in the earliest Greek mythology, the seven sisters in the Ursa Minor would be the Hesperides, daughters of Atlas. Together with other constellations near the zodiacal constellation of Scale (Boieiro, Ursa Major and Dragon), they may have formed the origin of the myth of the apes of the Hesperides, which is part of the Twelve Works of Heracles.