Miriel was the wife of Finwe and Mother of Feanor. She died in Year 1170 YT after giving birth to Feanor. So were there any Elven deaths that pre-date hers, or is she the first to die?
Although she was the first named elf we know died, she was not the very first; there were Elves who died early in their history, before they met the Valar and learned about the immortality of their souls, but their names are not recorded:
[I]n the days when the minds of the Eldalië were young, and not yet fully awake death among them seemed to differ little from the death of Men.
It was in Aman that they learned of Manwë that each fëa was imperishable within the life of Arda, and that its fate was to inhabit Arda to its end.
History of Middle-earth X Morgoth's Ring Part 2 "The Second Phase" Chapter 3: "Laws and Customs Among the Eldar" Of Death and the Severance of Fëa and Hröndo [> Hröa]
Míriel's death was such a shock to the Valar not because she died (the Valar knew the elves could be killed, though they didn't naturally die) but because she died in the Undying Lands1, where (in theory) there was nothing that could have caused her life to end prematurely. So she was the first in that regard, but not in a literal "first elf in all of history" sense.
1 There was also the matter of her (ex-)husband wanting to remarry after her death, which was also rather unprecedented
It appears she is the first named Elf to die out of all of the Elves, from Lotr wiki
After giving birth to Fëanor she could no longer live and so wished to die, something which was impossible because, as an elf, she was immortal within Arda. In YT 1170, she laid down in his gardens where her fëa peacefully departed from her body, and she entered the Halls of Mandos: in essence she had died of free will. Her body was tended by the maidens of Estë and remained preserved. This was seen as a shocking event by the Valar, and eventually led to the second marriage of Finwë,ote
from Tolkien's work this comes from
The Silmarillion, Quenta Silmarillion, Chapter VI: "Of Fëanor and the Unchaining of Melkor"
 The History of Middle-earth, Vol. 10: Morgoth's Ring, The Annals of Aman
The History of Middle-earth, Vol. 10: Morgoth's Ring, The Later Quenta Silmarillion, The Second Phase, IV: "Later Versions of the Story of Finwë and Míriel"
Miriel was not the first elf to die. She was probably not the first named elf to die either.
If you count an essay written by Tolkien, it gives a legend about the first elves to awake. It names the three leaders of the three groups and their wives. But when the elves were summoned to Aman their numbers had multiplied many times and their three kings had different names and had siblings and other relatives, and thus were probably second or later generation elves. Thus the first three named elf leaders were probably already dead by then, killed by various dangers of Middle-earth.
Miriel isn't the first elf to die.
In Miriel's case, she died because of the enormous amount of life energy drawn from her by her son Feanor during birth.
In YT 1170, she laid down in his gardens where her fëa peacefully departed from her body, and she entered the Halls of Mandos.
In the beginning, there were many nameless elves who had died. As quoted from History of Middle-Earth, Morgoth's Ring:
As ages passed the dominace of their fear ever increased, ’consuming’ their bodies (as has been noted. The end of this process is their ’fading’, as Men have called it; for the body becomes at last, as it were, a mere memory held by the fea; and that end has already been achieved in may regions of Middle-earth, so that the Elves are indeed deathless and may not be destoryed or changed. Thus it is that the further we go back in the histories, the more often do we read of the death of the Elves of old; and in the days when the minds of the Quendi were young and not yet fully awake death among them seemed to differ little fom the death of Men.
This was before their coming to Valinor and finding by Orome. These elves were not educated in their gift of immortality and thus threw away their lives as easily as Men did.
Therefore, you can say that Miriel was the first named and known elf to have died, but not the first elf in general.
In addition, it may have been possible that some elves had died of the torture by Melkor during their mutilating into orcs. It isn't exactly stated but one can assume so. From the silmarillion:
Melkor discovered the Elves before the other Valar, captured many of them, and transformed them by torture and other foul craft into orcs.