Warning: the following may contain spoilers for The Rings of PowerThe Rings of Power may have already revealed the origin of the three Elven rings. Despite the show's name, The Rings of Power hasn't actually gotten to the point where the titular rings leading to the events of The Lord of the Rings are actually created, but that could be about to change.
As is highlighted in the prologue to Peter Jackson's three Elven rings of power.
The Rings of Power's Mithril Plot is an Important Part of Sauron's Ring Plans
As told in The Silmarillion (which Amazon doesn't actually have the rights to, so the story may differ slightly), Sauron approaches the Elves as a fair figure named Annatar, the "Giver of Gifts." In this form he educates the Elves on the craft of ring-making, teaching them to forge rings of power. The Elven smiths are tricked into making many rings at Sauron's behest, but the three Elven Rings specified in the Ring Poem were created by Celebrimbor in secret, free of Sauron's influence, although they were still unknowingly bound to his One Ring. When Sauron puts the One Ring on his finger, the wielders of the three Elven rings, the seven Dwarven rings, and the nine rings of Men become subject to his will, although it appears as if the nine kings of Men were the only ring wielders to fully submit to Sauron's corruption.
In the first few episodes of The Rings of Power, it's revealed parts of this plan are already in motion, as Celebrimbor is already building mighty furnaces at Eregion with aims to craft something of "real power." Elrond is unknowingly pulled into this plot when they have him attempt to forge an alliance with the Dwarves to help build the towers (not knowing Celebrimbor and Gil-Galad hope to get the mithril, too), but it's likely Sauron, as Annatar, had already approached Celeborn, and may even be the one who stoked the Elves' fear of fading and stoked their hopes with the idea to use the Dwarves' mithril to restore the light of the Eldar.
The Elven Rings Solve Gil-Galad and Celebrimbor's Elf Mortality Fears
Conveniently, while the Elves' initial plan was to get a large supply of Mithril and use it to douse every Elf in Middle Earth in the Light of the Trees once again, the Elven rings of power could provide a different solution to the same problem, but require far less mithril. Very little is known about the three Elven rings of power, Narya, Nenya, and Vilya (at least compared to what we know about The One ring), but The Silmarillion specifies "those who had them in their keeping could ward off the decays of time and postpone the weariness of the world." It can't be overlooked that the powers of these rings explicitly provide the same solution The Rings of Powers' Elves are seeking with the Dwarven mithril.
In episode 7, "The Eye," King Durin refuses to allow Prince Durin to mine for more mithril and denies aid to the Elves. Celeborn and Gil-Galad aren't likely to accept their eventual mortality just because they can't use the mithril, so the Elves are likely to begin developing a new plan to "ward off the decays of time," which provides the perfect opening for Sauron, as Annatar, the Giver of Gifts, like the evilest of traveling salesman ready to show them how to craft rings of power that will ease all their anxieties.
Elrond's Mithril Could Be Used to Make Galadriel's Elven Ring of Power
Interestingly, while Vilya (eventually worn by Elrond) is a gold ring set with a blue sapphire and Narya (given to the Elf Lord Círdan, who eventually gives it to Gandalf) is a gold ring with a red stone, Nenya, the ring Galadriel eventually wields, is actually made of mithril. Since all three rings contain the power of preservation, it's obviously not a trait only inherent to Nenya's mithril, but the Elves' creating it to ward off the aging of their race, and the fact that it uses the same Dwarven metal Celeborn and Gil-Galad wanted to use for the same purpose, seems like more than a coincidence. Adding to the ring's significance, the show may have already revealed the mithril used in Nenya's construction.
King Durin forbid Prince Durin from mining any more mithril or sharing any mithril with the elves, both of which we know happen at some point in the timeline, but it doesn't seem like it'll be in time to solve the Elves' current concerns. Fortunately, the Elves won't be dependent on the Dwarves to provide any mithril to make Nenya since Elrond already has a piece of mithril containing more than enough of the metal to craft a ring. The piece of mithril Durin originally gave Elrond has been heavily featured in multiple episodes, and Elrond is highlighted holding it in a drawn-out moment following his expulsion from Khazad-dûm by King Durin.
Vilya and Nenya, the rings worn by Elrond and Galadriel, ultimately fulfill the Elves' desires by preserving both Elrond's home in Rivendell and Galadriel's home in Lothlórien. The Silmarillion as "thus it was that in two domains the bliss and beauty of the Elves remained still undiminished while that Age endured: in Imladris, and in Lothlórien...where the trees bore flowers of gold and no Orc or evil thing dared ever come." Despite the power of the Elven rings ultimately preserving the Elves' longevity as desired, their creation was also part of a plot crafted by Sauron that brought even more evil into the world, meaning The Rings of Power may have some dense thematic ideas for viewers to chew on in the coming seasons.
Considering the concept of mithril containing the light of the Silmarils and potentially providing a healing light to the Elves is entirely new lore appearing exclusively in The Rings of Power, it's hard to know exactly how this plot will resolve, but the motivation to eventually create the Elven rings has been firmly established. With the season finale approaching, we're sure to see big reveals surrounding Sauron's ring plot and maybe even find out if Elrond's small piece of mithril is related to the creation of Nenya.