Shalom Jonathan,
I do not really know. The “Ring of Solomon” is attached to an interesting story. Here is what I’ve heard:
One day King Solomon decided to humble Benaiah ben Yehoyada, his most trusted minister. He said to him, "Benaiah, there is a certain ring that I want you to bring to me. I wish to wear it for Succot, which gives you six months to find it."
"If it exists anywhere on earth, your majesty," replied Benaiah, "I will find it and bring it to you — but what makes the ring so special?"
"It has magic powers," answered the king. "If a happy man looks at it he becomes sad, and if a sad man looks at it he becomes happy." Solomon knew that no such ring existed in the world, but he wished to give his minister a little taste of humility.
Spring passed and then summer, and still Benaiah had no idea where he could find the ring. On the night before Succot he decided to take a walk in one of the poorest quarters of Jerusalem. He passed by a merchant who had begun to set out the day's wares on a shabby carpet. "Have you by any chance heard of a magic ring that makes the happy wearer forget his joy, and the broken-hearted wearer forget his sorrows?" asked Benaiah.
He watched the grandfather take a plain gold ring from his carpet and engrave something on it. When Benaiah read the words on the ring, his face broke out in a wide smile.
That night the entire city welcomed in the holiday of Succot with great festivity. "Well, my friend," said King Solomon, "have you found what I sent you after?" All the ministers laughed, and King Solomon also smiled.
To everyone's surprise, Benaiah held up a small gold ring and declared, "Here it is, your majesty!" As soon as King Solomon read the inscription, the smile vanished from his face. The jeweler had written three Hebrew letters on the gold band: “gimel” “zayin” “yud”, which are the first letters of the words “Gam Zeh Ya'avor”, which means "This too shall pass."
At that moment King Solomon realized that all his wisdom and fabulous wealth and tremendous power were but fleeting things, for one day he would be nothing but dust. King Solomon is famous as an extremely wise king, who prayed to God become filled with even greater wisdom. The humbling message of the humility with which a person should live — even a king — was a lesson that was intended to increase his wisdom, and the wisdom of all who would hear this story.
It's a beautiful story, but unfortunately there is no source to it as far as I am aware of. I think that it is likely a fable used by people trying to teach an important lesson in a way that which catch the listeners’ imaginations. But if anyone who reads this has more information about the ring or this story, please contact me at ask@askthetherabbi.org .
