Lady Stoneheart never appeared in Game of Thrones, but the ending of Sandor Clegane a.k.a. The Hound might tell us what's going to happen to her in George R.R. Martin's final two A Song of Ice and Fire books. The future of Lady Stoneheart remains one of the biggest mysteries to be resolved in The Winds of Winter and A Dream of Spring, but could she sacrifice herself in a similar fashion to The Hound?
Lady Stoneheart, the reanimated Catelyn Stark who is brought back to life by Beric Dondarrion, has only briefly appeared in Martin's novels, but already managed to make a big impression. The reveal of Lady Stoneheart in A Storm of Swords' epilogue is a jaw-dropping twist, and fans have been left waiting a long time to discover what will happen next between her and Brienne, whom she was ready to hang in A Feast For Crows. Because Catelyn returning from the dead as a scarred, zombie-esque being who desires only vengeance on the Lannisters, Boltons, and Freys is kind of a big deal, it was expected that Lady Stoneheart would appear in Game of Thrones.
When Lady Stoneheart didn't turn up in the Game of Thrones season 4 finale, hopes waned only slightly, but then increasingly so following the season 5 finale, but it was in season 6 when it really became apparent that Lady Stoneheart wasn't going to be included in Game of Thrones, much to Martin's chagrin. When the Brotherhood Without Banners returned, they were still being led by Lord Beric. The Hound then joined them too, even urinating in the river as if to hammer home the fact Catelyn wasn't coming back.
Lady Stoneheart's book role largely passed to her two daughters: at the end of "Battle of the Bastards", Sansa killed Ramsay and wiped out House Bolton; an episode later, Arya killed Walder Frey, and then completed the annihilation of House Frey in the season 7 premiere. Arya, in particular, was driven by a quest for vengeance, and it had been achieved in a satisfying way, meaning it would've been difficult to fit in Lady Stoneheart around that (not to mention Game of Thrones' Catelyn was a warmer, more sympathetic character than ASOIAF's).
Still, there needs to be more to Lady Stoneheart than just getting revenge on those who wronged her and her family, and that's where the Hound and his relationship to Arya come in. The Hound reunited with Arya in Game of Thrones season 8, with the pair traveling to King's Landing on their own personal revenge missions: the former to kill his brother, and the latter to kill the Queen. When they arrive in King's Landing, unfortunately timed to coincide with Daenerys' complete destruction of the city, only one of them goes through with the plan.
The Hound pushes on to fight the Mountain in Cleganebowl, but he stops Arya from following him, saying: "You think you wanted revenge a long time? I’ve been after it all my life. It’s all I care about. And look at me. Look at me! You want to be like me?" With the Red Keep crumbling around them, the Hound knows that Arya continuing down the path of vengeance would likely lead to her death there and then and, if by some miracle she did survive, it would warp her until she became like him. She thanks him, touchingly saying his first name for the first time.
The Hound does indeed die in battle with his brother, but because of his sacrifice, Arya is not only able to make it out alive, but also leave her quest for revenge behind. She moves on, becoming more peaceful with who she is and fully reconciling her identity as a Stark while being able to forge her own path. In the books, that's a role more likely to fall to Lady Stoneheart than it is Sandor Clegane.
It's Arya, while warged into Nymeria, who drags Catelyn's body from the river. That establishes a direct link between the two characters, as does the fact both are defined by a desire for vengeance on those who killed their family. But Catelyn is not someone to follow; her scarred visage and sheer lack of remorse don't make her a hero, but a warning sign. Arya's story is all about identity and revenge, and it's her mother who will be the one to finally make her reclaim one and forsake the other. The circumstances will obviously be different, since Catelyn won't be linked to Cleganebowl, but where in Game of Thrones it's the Hound who is finally able to turn Arya away from her kill list, it'll be Lady Stoneheart in the A Song of Ice and Fire books.