Here's what we know:
* Arvedui was the last king of Arnor. He was a powerful and wise ruler, but he was also very lonely and isolated.
* The Snowmen were a race of beings who lived in the far north. They were not evil, but they were also not truly friendly. They were more like a force of nature, indifferent to the affairs of men.
* Arvedui gave his ring to the Snowmen as a gesture of friendship. He hoped that they would remember him and his people.
* The ring itself was not magical. It was simply a token of Arvedui's generosity.
It's important to note that this is all based on Tolkien's writings. The precise nature and significance of the ring is open to interpretation. However, there is no evidence to suggest that it was the One Ring.