Lúthien was a princess. She was born of an Unbegotten and a Maia under the brightest Stars crafted by Elbereth, whom her mother Melian knew as the Valaië Varda. Her mother had married the stubborn, stiff-necked Teleri-Sinda King who was her father and Melian was gifted with foresight.
A foresight which she passed unto her daughter. Lúthien was well aware of what her future might hold, and in another world, she might have happily met her fate, might have happily felt love for the mortal Beren Ercharmion. But she was not that Lúthien. She was stronger, more stubborn, and she knew what could ultimately bring their world from the darkness of the Black Enemy of The World.
She would fight it herself, not leave future generations to continue what she had the power to undo. Her mother would not lose her daughter to death beyond the circles of the world, and Lúthien knew enough of the future to become the mother of important figures. She was half-elven, she could choose when to birth a child, and she was also half-maiar, able to Hear the Song. She could choose the souls which she would bring into the world, she could raise them properly, without quite so much grief and loss.
She might be severely disappointing her parents, but Lúthien was strong enough to change her own fate in one future, she was surely strong enough to change it, and therefore her descendants, in another.
In one life, Lúthien Tinúviel chose to love a mortal man named Beren, and through him—after stealing a Silmaril from Morgoth's crown and battling Sauron— she bore a child named Díor Eluchíl, who bore a daughter named Elwing (as well as twin sons), who bore twin boys named Elrond and Elros.
In another life, Lúthien Tinúviel did not marry Beren Erchamion, but still stole the Silmaril from Morgoth's crown. She did not lead Finrod to his death—though he found it still—and she chose to bear twin sons whom she named Elrond and Elros, but their Maia blood was not nearly so diluted as it could have been, and they bore no trace of Man. Elros still chose to align his fate with a mortal woman, which did sadden his mother, but Elrond was one-quarter Maia, and far more capable of fighting back the shadow through the Ages. He also had his Sindarin mother and Noldorin father for far longer.
In one world, Lúthien, daughter of Elu Thingol and Melian the Maia, chose to die for a mortal man; in another, she chose to live for herself. For her future.
This both changes things and also changes nothing.
The idea is that she marries a Fëanorion—probably Maglor, and gives him the Silmaril. When the surviving sons of Fëanor touch said Silmaril—freely given, they are released from the oath regarding said Silmaril (Maglor completely due to his marriage bond somehow).
Lúthien bears Elrond and Elros (skipping Díor and Elwing) in the line to the Fëanorion. Her father still demands the Silmaril for her bride price (after she the sons of Fëanor touch said jewel and are released from its hold (one of three).
The dwarves still attack Doriath, and Lúthien is with Maglor, not in a forest near Menegroth. Elu Thingol still dies and Melian still flees. But the repercussions of that single sacking of Doriath are worse than they would have been had Lúthien already passed from the circles of the world. Most who died in the second kinslaying die during the attack by dwarves—though they also end up slaughtering the dwarves in revenge.
Celeborn and Galadriel, Nimloth, and Orophers family escape as they would have, and maybe a few others that would not have(in cannon) as well.
The "Freed" Silmaril is taken from Menegroth and eventually someone ends up taking it (Eärendil and whoever becomes his wife—some cousin to Elrond and Elros?) to the Valar in Aman to ask for aid against the Morgoth.
There is no Third Kinslaying, only slaying of the enemy (though the sons of Fëanor who would have perished in the Havens of Sirion perish in battle against Morgoth's forces instead).
Gondolin still falls.
The War of Wrath still happens. And it ends nearly the same way. Though the other two Silmarilli are stolen, though without another kinslaying and Lúthien's husband is not part of it at all, though he somehow ends up being the only Fëanorion still alive anyway.
Elros, who has fallen for a mortal woman, renounces his fate—much like Lúthien once did—and becomes the first King of Númenorë.
And so the First Age of the Sun ends. Lúthien and her Fëanorion spouse are still alive, Elrond and Elros have chosen the same fates they would have otherwise, and Ereinion Gil-Galad is still High-King of the Noldor due to his Fëanorion cousin having long given up on any claim.
Lúthien does not take over for the Sindarin people, instead leaving that to her kin Oropher ect, and drops out of history with her spouse…mostly, though they still speak with Elrond (and Elros before he quits Arda).
Then…mostly cannon-ish from there?
Her father is strong and full of Power and her mother is Melian the foresighted Maia.
Lúthien inherits both strength of will and Power. She knows possible futures and is NOT content to die a mortal death.
Lúthien does not fall for Beren Erchamion, however pleasant the man might possibly be. Beren does not go on a quest to steal a Silmaril, but Lúthien does, and she succeeds much like she might have for love of Beren, but instead it is for love of Middle-Earth.
Lúthien lives and changes everything.
Lútien lives and doesn't change much at all.