Speculation: Changing Perrin's Story in the Two Rivers
So I have a pretty big theory about what's going to happen to TV Perrin that's 100% not in the books. If they don't do it, I won't be surprised because it is pretty out there, but it'll certainly create waves if true.
Spoilers up to Book 6, as well as wild speculation about Season 3
TLDR: Alanna is going to symbolically take Berelain's spot as the Hawk, and she will forcibly bond Perrin instead of Rand as her Warder.
I think this because of the timing of Alanna losing Ihvon, seemingly in the Tower Battle before eventually going to the Two Rivers, where Maxim is seen alive and leading the battle. The scene with Alanna walking up to Perrin with the cherry blossoms in the background gave me dangerous vibes, especially because I don't remember them being that important to each other in the books.
In short, thematically it's about dealing with grief: Alanna losing Ihvon can mirror TV Perrin mourning Laila, especially since returning to the Two Rivers would exacerbate his feelings. In the books, Perrin loses his whole family offscreen in Book 4, but the TV show propped up Laila to be his doomed wife for this same tension point. Plus there's a moment in the latest trailer where Perrin and Maxim share a meaningful look. Possibly the warder bond? And Maxim would be grieving too since he was also in love with Ihvon.
Also, Faile being around can create a more believable wedge between the Hawk and the Falcon than in the books, especially if Alanna succeeds in bonding Perrin. I always felt the dynamic between Berelain and Faile was strained, so if I'm right about my guess, then that's one messy plotline out of the way. Instead of Faile being defensive of the possibility Berelain stealing Perrin away, Alanna already has that connection to him. And it's just as terrible as it was in the books (possibly more so).
Plus Alanna's actress is lovely and a boon for the TV series, so this change can keep her and Maxim relevant longer. Also the white face paint might be another sign of her mourning Ihvon, since it's not there when she's screaming in grief in the shot from both s3 trailers.
However, it goes without saying this is a big change from the books. Alanna bonding Rand against his will in Lord of Chaos was a shock to me and really ramped up his distrust of Aes Sedai. But beyond that, I don't know if it had major repercussions in the story. (Correct me if I'm forgetting something) Besides, there's plenty of other Aes Sedai tomfoolery to piss him off, so I'd argue it's not completely necessary to begin with. Also it seems less meaningful for Alanna to bond Perrin instead of Rand, but playing her up as the Hawk would fix that IMO
Moreover, all this could add more tangible drama for the TV show, compared to the quieter internal struggle Perrin has throughout the book series. Overall I'm for it, since Perrin is my favorite character and I live for The Drama. It takes an arguably forgotten plot element from the books and gives it a satisfying knife twist to keep at least me invested.
Anyway since this is a big departure from the books, I'm not 100% confident it will happen... And I'll know I'm wrong if they finally introduce Berelain to the show. But if it DOES happen, well... You read it here first 🥇