We have no idea, but in some earlier drafts and notes it was said to be for "an elf-king’s son" or for the sons of "Girion king/Lord of Dale".
Tolkien never gives any indication as to which specific elf the mithril coat was made for. We can of course speculate random characters, but it could have just as likely been a character who's not mentioned.
There's not a lot else to say about that idea, so let's take a look at the manuscript history of this passage and see what we do know.
We can trace the development of what would become the mithril coat through some of Tolkien's plot notes and drafts for the Hobbit. (Do note though that all of this long predates the significance of the coat being of Mithril. That detail was invented for The Lord of the Rings, and only added to The Hobbit through a revision to the text in 1966.)
In some plot notes Bilbo dons a suit of mail that was made for "an elf-king’s son" (who does not seem to be the king of the wood-elves in the story).
With an army – a battle is gathering in the west. B[ilbo] puts on the a suit of silver mail made for an elf-king’s son, and goes with the wood-elves to battle.
The History of the Hobbit - The Second Phase, Plot Notes B
In some plots written a bit later, this idea drops, but not now we instead get told of silver mail suits made for Girion's sons. (Girion's jem is the precursor for the Arkenstone.)
B[ilbo] goes back and talks to dwarves. Warns them dragon knows of exit. Asks them about plans. They are a bit flummoxed. They tell him of the Jem of Girion king of Dale, which he paid for his sons’ arming in gold & silver mail made like steel.
The History of the Hobbit - The Second Phase, Plot Notes C
These coats for Girion's sons then make their first narrative appearance in the first manuscript draft of the "Conversations with Smaug" chapter, (written right after those notes), as something the dwarves tell Bilbo about.
There they sat and the talk drifted on to things they remembered, that must
now be lying in the hall below ....and most fair of all the white gem of Girion Lord of Dale, which he paid for the arming of his sons, in coats of dwarf mail the like of which had never before been made of silver wrought the power and strength of steel
The History of the Hobbit - The Second Phase, Chapter XII
And then in the first manuscript draft of the "Not At Home" chapter, we actually see such a coat, though this time one made for "some elf-prince", in what would eventually become the passage we're familiar with.
Then he put upon Bilbo a small coat of mail, wrought for some elf-prince long ago. It was of silvered steel, adorned with pearls, and a belt of pearls and crystals went with it.
The History of the Hobbit - The Second Phase, Chapter XIV
At some point later Tolkien penciled in a change of "some elf-prince" to "some young elf-prince". (John Rateliff suggests that this was because when Tolkien originally wrote it he was envisioning his elves as smaller than human sized, but then by making the elves taller, for the coat to still fit Bilbo Tolkien specified that it was for a young elf.)
This is possibly of significance, because it suggests that Tolkien might have conceived of the elves as somewhat smaller than human size when he originally wrote this passage. Initially, in his early ‘fairy poetry’ such as ‘Goblin Feet’ and ‘Tinfang Warble’ and in The Book of Lost Tales, Tolkien had thought of the elves as much smaller than human, but by the mid-1920s came to reject this and envisioned them instead as of similar stature to humans (as in the feys of medieval romance, legends of the Tuatha dé Danaan, and Spenser’s Faerie Queene).
The History of the Hobbit - The Second Phase, Chapter XIV, note 16
And then as first published in 1937, the passage read:
With that he put on Bilbo a small coat of mail, wrought for some young elf-prince long ago. It was of silvered steel and ornamented with pearls, and with it went a belt of pearls and crystals.
The Annotated Hobbit