Game of Thrones' White Walkers were set up to be the major villains of the story, and their motivations were finally explained in season 8, but George R.R. Martin likely has some very different plans for them in the books. Martin still has two A Song of Ice and Fire novels to write, with Game of Thrones ending before the books.
After breaking through the Wall in the season 7 finale, it appeared as though the battle with the White Walkers, led by the Night King, was to be the major arc for Game of Thrones season 8, with his motives a mystery to be solved. As revealed ahead of "The Long Night", the Night King's target was Bran Stark, or rather, the Three-Eyed Raven. His goal was to erase Westeros, and that started with the Three-Eyed Raven, who served its memory. In order to wipe out humankind, he needs to destroy the keeper of its stories. Fans had long wondered what exactly the Night King and White Walkers wanted, and weren't completely satisfied with the result - not least because they were defeated so soon afterward.
In Martin's books, things are likely to go down rather differently. The White Walkers in the novels aren't led by the Night King, and although they're intelligent and perhaps have a hierarchy, they're unlikely to be so singularly focused on the Three-Eyed Crow. Instead, we can expect the Others to serve more as a demonic force against Westeros that wants to lay waste to the living; in short, the White Walkers are death made flesh (or at least ice), and the humans all need to band together to defeat them. This ties into some of what we saw in Game of Thrones prior to the Night King's motivation being revealed.
Because Martin has always tried to subvert most fantasy tropes, there have been numerous theories attempting to explain what the White Walkers want. It's said that they've returned because of dragons; that they're linked to the Starks; or that they're the servant of the Great Other, a god who is the enemy of the Lord of Light. Another popular theory is that they represent climate change - a parallel Martin himself admits is there - and while it's unlikely that's what the author was thinking about when developing his story back in the 90s, it does provide the closest basis for what the White Walkers do actually want.
There likely will be some more to the White Walkers in the books. It would be interesting if Martin devoted some time to the Land of Always Winter and explored some of that story, and there could be grander reveals about the Children of the Forest's own objectives. But since the White Walkers are an inhuman, demonic race, then they shouldn't be motivated by revenge or wanting to erase a specific person. Instead, they're part of Martin's endpoint for his expression of Samuel Faulkner's belief that "the human heart in conflict with itself is the only thing worth writing about."
That's the core of Game of Thrones and A Song of Ice and Fire, and is why the White Walkers really exist: they're a terrifying force of nature, and all of the different characters need to band together and fight them, or die. That's why they're the ultimate test. Human characters can be evil, but the Others just are. The Long Night in the books will be different (and much, much longer), tying into various prophecies and serving different character arcs (such as coming after Daenerys' attack on King's Landing), and the motive won't be exactly the same as on Game of Thrones either. But rather than being some convoluted explanation, it's more likely that the White Walkers just want to wipe out existence itself.