What was the white gem Frodo wore?

Question

In the Return of the King, the last chapter "The Grey Havens", Frodo appears to have a white gem hanging from a chain around his neck that is mentioned a couple of times:

On the thirteenth of that month Farmer Cotton found Frodo lying on his bed; he was clutching a white gem that hung on a chain about his neck and he seemed half in a dream.

And:

Frodo and Sam, however, went back to ordinary attire, except that when there was need they both wore long grey cloaks, finely woven and clasped at the throat with beautiful brooches; and Mr. Frodo wore always a white jewel on a chain that he would often finger.

In both cases he seems to treat it like a proxy for the One Ring; the first quote seems to be some kind of flashback/despair for the Ring being destroyed, and in the latter quote he "would often finger" it, implying that it relates to the Ring.

What is this white gem/jewel? At first I wondered if it could have been the Phial of Galadriel, but he gave that to Sam, and it's not a gem or jewel either, so it's not that.

Answer 1

The white gem given to him by Arwen

Arwen, in the chapter "Many Partings" had given Frodo a white gem to help him with his hurts and pains. It is this gem that Frodo continues to wear and that is seen above. This is one of the very few mentions of the gem, most of which give no more description than below:

But the Queen Arwen said: ‘A gift I will give you. For I am the daughter of Elrond ... But in my stead you shall go, Ring-bearer, when the day comes, and if you then desire it. ... But wear this now in memory of Elfstone and Evenstar with whom your life has been woven!’... And she took a white gem like a star that lay upon her breast hanging upon a silver chain, and she set the chain about Frodo’s neck. ‘When the memory of the fear and darkness troubles you,’ she said, ‘this will bring you aid.’
_{Return of the King - Book VI, Chapter 6: Many Partings}

One of the few mentions in earlier versions of the Legendarium was that the gem would serve almost as a ticket to the ship to the West, suggesting that with the gem, Elrond would not refuse him. This was, however, immediately rejected and replaced with taking the Phial of Galadriel. The tone seems to suggest there would be no reason for Frodo to need anything for Elrond to recognise him. It is also clear from the below, that this gem is not the Phial of Galadriel, as that is mentioned seprately.

'... But in my stead you shall go, Ringbearer, when the time comes, and if you then desire it: for your wounds have been grievous and your burden heavy. But you shall pass into the West until all your wounds and weariness are healed. [Struck out at once: Take this token and Elrond will not refuse you.' And she took from her hair a white gem like a star] Take with you the Phial of Galadriel and Cirdan will not refuse you. But wear this now in memory of Elfstone and Evenstar with whom your life has been woven!' And she took a white gem...
_{History of Middle-earth - Volume 9, Sauron Defeated - Part One: The End of the Third Age, Chapter VII: Many Partings}

Answer 2

Queen Arwen had "a white gem like a star that lay upon her breast hanging upon a silver chain". Before Frodo returned home, she gave Frodo a white gem -- probably that same one:

But the Queen Arwen said: ‘A gift I will give you. For I am the daughter of Elrond. I shall not go with him now when he departs to the Havens: for mine is a choice of Luthien, and as she, so have I chosen both the sweet and the bitter. But in my stead you shall go, Ring-bearer, when the day comes, and if you then desire it. If your hurts grieve you, and the memory of your burden is heavy, then you may pass into the West, until all your wounds and weariness are healed. But wear this now in memory of Elfstone and Evenstar with whom your life has been woven!’

And she took a white gem like a star that lay upon her breast hanging upon a silver chain, and she set the chain about Frodo’s neck. ‘When the memory of the fear and darkness troubles you,’ she said, ‘this will bring you aid.’