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.
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.