Queen Elizabeth I owned a prize possesion that was thought to be a rare natural item of magic. What was this item, who gave it to her, and what really was it?

You could be talking about bezoar stones. Queen Elizabeth had a few of them seemingly. I can't find who specifically gave them to her. They have been valued for centuries as rare magical things that counteract poisoning and plenty of other nonsense. Unfortunately, they are just stones that are obtained from the livers of animals. In other words, gallstones.

https://www.wondersandmarvels.com/2010/11/royal-sparkly-things.html

http://medical-dictionary.thefreedictionary.com/gallstone

OK, I think it may be a unicorn horn. This was given to her by Martin Frobisher, a captain in the navy. The Unicorn's horn turns out to be a narwhal's tooth.

http://onthetudortrail.com/Blog/2010/10/20/elizabeth-is-magical-unicorn-by-carlyn-beccia/

Outstanding!

Probably the most famous alicorn of all—known as the Horn of Windsor—belonged to Elizabeth I of England. The horn was listed among Elizabeth's crown jewels and valued at 10,000 pounds (more than 10,000,000 pounds at today's prices), a sum which at that time would have been enough to buy a large estate plus castle.

This horn was not given to the queen as a gift from another monarch, but came directly from the man who found it—Martin Frobisher. A captain in the British Navy, he had been trying to discover a northwest passage to India for some time. During his first attempt in 1576, rough winds and cold weather forced him to turn back. But the trip was not a total failure—some of his men had found some "black earth" and the rumor quickly spread it was gold. This made it much easier for him to find backers for future journeys and he was able to set out again the very next year. Once more inclement weather interfered with his explorations. And, after several ships were wrecked by a storm, Captain Frobisher decided to end his journey. He had sailed as far as the inlet now known as Frobisher's Bay in Baffin Island, Canada. His men, who spent most of their time there collecting ore, found "a great dead fish" with a hollow spiraling tusk almost two yards long. During the Middle Ages and the Renaissance, many people believed that for every animal of the land there was an equivalent animal of the ocean—Captain Frobisher and his men probably believed this animal was a sea unicorn. The sailors even tested the horn by placing poisonous spiders in the inner cavity. When they died, this provided adequate proof that the horn must belong to the unicorn of the sea. Frobisher returned to England and delivered the horn to Queen Elizabeth. He was later knighted for his valor against the Spanish Armada.

[unicornlady.net/treasures/treasures.html, no longer online]