Robert Baratheon put it there
Back in the very first episode, "Winter is Coming", Robert asks Ned to take him to see Lyanna. When down in the crypts he puts a feather in the hands of Lyanna's statue and this appears to be the same one.
You can see it in this scene:
The annotated script on Genius also makes note of it.
[Scene changes to the Winterfell crypt, at Lyanna’s tomb. Robert places a feather in the hand of her statue.]
ROBERT: Did you have to bury her in a place like this? She should be on a hill somewhere with the sun and the clouds above her.
NED: She was my sister. This is where she belongs.
Game of Thrones, Season 1 Episode 1, "Winter is Coming"
It's also worth noting that back in Season 5 Episode 4, "Sons of the Harpy", we see Sansa pick the feather up and dust it off. I don't think we see her put it back on the statue though but we can infer that she did. The scene can be seen here:
As for its significance, well it appears as though it is hinting at Jon's true parentage and is also symbolism for Robert's doomed love for Lyanna. First off Jon knocks the feather off the statue as he is the son of Lyanna and Rhaegar which broke Robert and at the same time a voice over of says:
Lyanna: You have to protect him.
Game of Thrones, Season 8 teaser/trailer
This echoes what Lyanna says to Ned when she hands over the baby Jon in the flashback scene at the Tower of Joy.
The feather then freezes over which further references the Robert's doomed love but could also be a reference to the tourney at Harrenhal, which Petyr and Sansa discuss in the before mentioned scene. Here they tell of the "crown of winter roses in Lyanna's lap, blue with frost" that Rhaegar lay in Lyanna's lap.
BAELISH: I saw her once. I was a boy, living with your mother's family. Lord Whent had a great tourney at Harrenhal. Everyone was there. The Mad King, your father, Robert Baratheon. And Lyanna, she was already promised to Robert. You can imagine what it was like for me, a boy from nowhere, with nothing to his name, watching these legendary men, tilting at the lists. The last two riders were Barristan Selmy and Rhaegar Targaryen. When Rhaegar won, everyone cheered for the prince. I remember the girls laughing when he took off his helmet and they saw that silver hair. How handsome he was. Until he rode right past his wife, Elia Martell, and all the smiles died. I've never seen so many people so quiet. He rode past his wife, and he lay a crown of winter roses in Lyanna's lap. Blue as frost. How many tens of thousands had to die because Rhaegar chose your aunt?
SANSA: Yes, he chose her… and then he kidnapped her and raped her.
Game of Thrones, Season 5 Episode 4, "Sons of the Harpy"
Note that I've seen reference to Robert having given Lyanna feathers as a symbol of his love before riding off for the Rebellion but this appears to be untrue as I can find no canon reference to this myself.
As Jon walks by, a feather falls from the statue. This is the feather that King Robert put there in the Season 1 pilot, when he visited the crypts during his stay in Winterfell. Robert would bring feathers to Lyanna before the rebellion as a symbol of his love, and Sansa finds that final feather still in the crypts in Season 5, lying at Lyanna’s feet, and returns it to the statue’s hand.
The Ringer, What the New ‘Game of Thrones’ Trailer Tells Us About Season 8’s Key Themes
This article also makes note of Robert doing this in the none canon pilot episode rather then the very first episode where it does happen as the released script for the actual pilot doesn't include the feather. Therefore, I'd say to be careful to take this article at face value.
