And indeed they did. In the midst of the battle -- just after Beric has perished -- Melisandre emerges and sees Arya and The Hound. The two women acknowledge that they know one another. "You said I would shut many eyes for ever," Arya says. "You were right about that too." To which Melisandre responds: "Brown eyes, green eyes... and blue eyes." She mirrors her original statement from way back, but emphasizes the "blue." In other words, she tells Arya: You're going to kill a bunch of White Walkers. However, it's not just that Arya will kill some of the merciless villains. Instead, she'll kill all of them.
Melisandre had long been touting the Azor Ahai prophecy, which stated that "only the prince who was promised can bring the dawn" during the Long Night. For years, the likely candidate for said prince was likely Daenerys or Jon. But now Arya emerges as a strong contender. "Prince" in Valyrian could mean man or woman, for one thing, and Arya has constantly disguised herself across genders and identities. She has been hidden as a boy and lived as a girl with no name. But now she's Ayra Stark, killer of the Night King. This seems like a pretty clear statement, especially combined with Melisandre's death, that Arya is the fulfillment of the Azor Ahai prophecy, though there are still some corners of the internet that disagree.
Of course, there are still lingering questions. For one: How did Arya manage to get to the godswood? There was a moment, lost to many viewers during the heat of battle, when she dashes out of the room in which she, The Hound, and Melisandre are hiding. That still leaves a long way to go to get to Bran in time, but Game of Thrones is not a show known for its stringent adherence to laws of time and geography (see: the ability of characters to travel all over Westeros in no time at all during Season 7).
More importantly, what comes next? The last two episodes have belonged to Maisie Williams' character. First, we see Arya evolve emotionally in her surprising sex scene with Gendry, and then her years training to become an expert assassin made her the perfect person to stop an icy apocalypse. With Cersei and her army preparing for the remnants of the living, nothing is off the table. Maybe Arya Stark's next stop is the Iron Throne.