Brienne of Tarth is one of the most surprisingly resilient characters on Game of Thrones, especially considering how beloved she is by fans. (As die-hard GoT-ers know, the more we love someone, the more likely they are to be taken from us in a particularly heinous way.) The fact that she has lasted this long, and easily killed off some of the more brutal villains on the show, gives us a tiny glimmer of hope that the one reason why Brienne will kill Ramsay Bolton on Game of Thrones will be enough motivation to add to her kill list.
Like The Hound and Stannis Baratheon, Ramsay Bolton has committed significant atrocities upon innocent young girls, namely one of the Stark daughters Brienne swore an oath to Katelyn Stark to protect. She has finally reunited with Sansa and pledged to protect her, which is not an empty promise, considering Brienne's considerable prowess in a sword fight. And as Sansa's torturer, Ramsay Bolton — a fairly useless fighter who stoops to sociopathy to defeat his enemies — is a prime target for Brienne.
Sansa and Jon Snow received word from Ramsay in the Season 6 episode "Book of the Stranger" that he has their little brother Rickon imprisoned in his dungeon. He beckons them to return to Winterfell to rescue him, promising to defeat them all, murder Jon and Rickon, and continue to torture Sansa before forcing her to bear him an heir. But Ramsay doesn't know that Sansa is packing heat in the form of a formidable and fiercely loyal warrior.
Ramsay has an army of 5,000 ready to fight, and Jon Snow has merely 2,000 Wildlings, who don't even necessarily serve him. Since he's abandoned his post as Lord Commander of The Night's Watch, he can't count on their help either — and even if he could, they conspired to murder him, so maybe they're not the best men to count on. Sansa points out that The North is loyal to the Starks, and if Jon asked them to rise up against Ramsay Bolton, they would join him in the fight. With Jon Snow leading an army that can hold off Bolton's men, Brienne has the opportunity to infiltrate the castle and finish off Ramsay herself — it's not like Ramsay is going to put himself anywhere near harm's way on the battlefield anyway.
It would certainly be one of Brienne's most triumphant moments on the show to rescue so many Stark children from the terror of Ramsay Bolton.