Ursa Minor
Ursa Minor is modernly known as 'The Little Dipper' the smaller constellation related to the Big Dipper. These two constellations share a lot with one another in terms of appearance, navigation and myths. The 1st star, Polaris, is the North Star and using the more obvious Big Dipper can be located easily.
Greek/Roman legend
Ursa Minor is usually associated with two different myths. In one, the constellation represents Ida, the nymph who took care of Zeus on the island of Crete when he was small, along with Adrasteia, the nymph represented by the larger constellation Ursa Major. Zeus’ mother Rhea hid Zeus on the island when he was very young to protect him from his father Cronus. Cronus, fearful of an old prophecy that said that one of his children would overthrow him, swallowed five of his children after they were born. When Zeus was born, Rhea tricked Cronus into swallowing a stone instead, and Zeus eventually fulfilled the prophecy. He freed his brothers Poseidon and Hades and sisters Hera, Hestia and Demeter, and became the supreme god of the Olympians.
In a different myth, the constellation represents Arcas, son of Zeus and the nymph Callisto. Callisto had sworn a vow of chastity to Artemis, but was later unable to resist Zeus’ advances and the two had a child, Arcas. When Zeus’ wife Hera found out about the betrayal and the child, she turned the nymph into a bear. Callisto spent the next 15 years wandering in the woods and avoiding hunters.
https://www.constellation-guide.com/constellation-list/ursa-minor-constellation/
Arcas was the son of Callisto, who was transformed by Juno into a bear. When Arcas was fifteen, he was out hunting in the forest when he came across a bear. The bear behaved quite strangely, looking him in the eyes. He of course could not recognize his mother in her strange shape, and was preparing to shoot her when Jupiter prevented him. Arcas was transformed into a bear like his mother, and the two were taken up into the sky. Juno was annoyed that the pair should be given such honor, and took her revenge by convincing Poseidon to forbid them from bathing in the sea. It is for this reason that Ursa Major and Ursa Minor are both circumpolar constellations, never dipping beneath the horizon when viewed from northern latitudes.
Conclusion
This mythical history of the Little Dipper or the Lesser Bear appears very shallow with other cultures other than the greek/roman legends. If the Little Dipper is anything other than Arcas or what seems a much earlier Ida, it was only used for the North Star, that most likely be a much more important star for navigation rather than storytelling. If the Little Dipper was Arcas, it wouldn't make much sense storytelling wise to punish him as well as his mother for never being able to bathe in the seas of Earth. So maybe it might not be a punishment for Ida but placing the protector of the god Zeus in the centre of the sky to always been seen and recognised.