Why Elrond Did Not Keep Heirlooms

[Arothir]

So one thing I always found interesting is that Elrond kept no heirlooms from his parents, with Elros seeming to take them all: Aranruth, Dramborleg, the Ring of Barahir, and the Bow of Bregor. This is seen again when he does not claim Glamdring when Gandalf brings it to Rivendell, as one of the two valid heirs for it (the other being Aragorn). For the latter perhaps Tolkien did not intend on Elrond having a claim on it or perhaps as its finder Gandalf was entitled to it? Was there anything written as to why Elrond really kept nothing from his parents?

[Lirimaer]

*follows* Oh this is an interesting premise. I have my own assumptions, but I look forward to seeing what is dug up.

[Mojo]

*resists the urge to make a pun about Elves weaving things out of their own hair*

Given that the Elves are immortal, my guess would be that he was able to preserve the memories of his parents and the history of his people. On the other hand, with Elros choosing to be mortal, heirlooms or "pieces of memory" needed to be physical items that could be passed down from generation to generation to preserve the beauty and memory of the Elder Days.

[Arothir]

The notion of memory makes sense, especially since he was planning on seeing them again when he went West. My only issue with that is that Elrond never met Bregor, Tuor, Thingol, or Barahir. Or perhaps maybe Elrond kept something, but it wasn't noted due to him not marching on Valinor and losing everything in the ocean?

[Mojo]

My lore is a little rusty, but did Elrond have a book that preserved the history of the Elves of Beleriand in the Elder Days? If not, there were certainly enough Elves in Imladris to preserve some artifacts of those days, though we aren't told about them much.

[Flame Fried Ent]

Besides, Elrond and Elros were raised by Maeglor, not their parents. Maybe Elrond has something of his? Beyond that, I agree with @Mojo about memories of Elves vs Men. Maybe, also, someone had the foresight to see that perhaps one day there might come a time when those heirlooms would be important to recognize and confirm the "real" King or ruler?

[Eldy Dunami]

My lore is a little rusty, but did Elrond have a book that preserved the history of the Elves of Beleriand in the Elder Days? If not, there were certainly enough Elves in Imladris to preserve some artifacts of those days, though we aren't told about them much.

You might be thinking of the account of Bilbo's Translations from the Elvish (given in the "Note on the Shire Records" in the Prologue to LOTR), which is stated* to be "a work of great skill and learning in which [Bilbo] had used all the sources available to him in Rivendell, both living and written. But since they were little used by Frodo, being almost entirely concerned with the Elder Days, no more is said of them here." There's is also a comment during the Council of Elrond that Elrond's full recounting of the history of the Ring is not given because "that history is elsewhere recounted, even as Elrond himself set it down in his books of lore" (FOTR, II 2).

*This statement was one of several changes to the framing device of LOTR between the first and second editions. Originally, Bilbo's work was not about the First Age, and we instead hear of "many annals, genealogies, and traditions of the realms of the South and the North [Gondor and Arnor], derived through Bilbo from the books of lore in Rivendell" among other sources (note prefixed to Appendix A; quoted in LOTR: A Reader's Companion, p. 682). Such material is still included in the version of the Red Book from the second edition, but Bilbo was evidently not involved in its preservation.

However, I would also draw attention to the following quote from "The Shibboleth of Fëanor" about the preservation of Elvish lore in the First Age:

All peace and all strongholds were at last destroyed by Morgoth; but if any wonder how any lore and treasure were preserved from ruin, it may be answered: of the treasure little was preserved, and the loss of things of beauty great and small is incalculable; but the lore of the Eldar did not depend on perishable records, being stored in the vast houses of their minds. When the Eldar made records in written form, even those that to use would seem voluminous, they did only summarise, as it were, for the use of others whose lore was maybe in other fields of knowledge*, matters which were kept for ever undimmed in intricate detail in their minds.

This is followed by author note 24:

And as some insurance against their own death. For books were made only in strong places at a time when death in battle was likely to befall any of the Eldar, but it was not yet believed Morgoth could ever capture or destroy their fortresses.

I think it would be reasonable to surmise that the Eldar of the Second and Third Age had a different attitude towards preservation of lore than in the First Age, and this may account for Elrond apparently having made written records of Second and Third Age history.* However, as late as the end of the Third Age (Bilbo worked on his Translation from the Elvish after leaving the Shire on his 111th birthday), oral records were evidently still an important source of information about Eldarin history, at least for the First Age (but, I would presume, later Ages as well).

*ETA: It slipped my mind initially, but this fits nicely with the account of Pengolodh given in "Quendi and Eldar", in which "before the overthrow of Morgoth and the ruin of Beleriand, he collected much material among the survivors of the wars at Sirion's Mouth concerning languages and the gesture-system with which, owing to the isolation of Gondolin, he had not before had any direct acquaintance. Pengolodh is stated to have remained in Middle-earth far into the Second Age for the furtherance of his enquiries" before leaving due to Sauron's invasion of Eriador (HoMe XI).