Since the universe-specifics of both Lord of the Rings and Game of Thrones are so similar, it would be tempting to assume that characters from either franchise would instantly become friends (or in some instances, instantly become enemies).
That said, it is certainly easy to image cross-platform friendships between certain characters — depending on how well they align with each other in terms of personality, ambition, or even motivation. On the other hand, there are a few pairings between LotR and GoT that would be extremely improbable for the same reasons.
Would Work: Frodo & Tyrion
Tyrion, having being reviled by his father, shunned by his sister, and only occasionally acknowledged by his brother, is sure to understand the burden that Frodo bears. As someone who has himself been the sole carrier of blame for everything that goes wrong with Westeros, it isn't hard for him at all, to be honest.
Although Frodo's job is not to create balance as much as it is to destroy the cause of disharmony, the One Ring, it is quite similar to the amount of backbreaking effort Tyrion takes to save the world from falling into peril. Best friends? Probably not, but drinking buddies at the very least.
Would Turn Ugly: Saruman & Yara
If there is one thing Yara Greyjoy cannot stand, it is people who are unable to handle the responsibility given to them, à la her brother, Theon.
This could just as well apply to Saruman, who fails to perform the one task for which he leaves Valinor (and instead tries to seize the very power he has been sent to repress.) In theory, the White Wizard's ambitions are quite similar to that of the Ironborn heir, but their methods of accomplishment are at opposing ends.
Would Work: Sauron & Ramsay
As much as Ramsay adores being the king in the castle, so to speak, what he loves more is lording his power and violence over other people, especially those who are powerless to fight back, which is a set of traits that pairs well with the Dark Lord's unexplainable lust for pointless brutality and torture chambers.
Even though one would think that Ramsay might resist serving under Sauron, he really wouldn't — in fact, it's likely that he'd be promoted to the next leader of Ringwraiths or something similarly macabre.
Would Turn Ugly: Cersei & Gimli
There would be a near-zero chance of these two characters becoming friends in any scenario, and not just because Cersei prefers the safety of the Red Keep while Gimli habitually thrusts himself into the midst of battle.
He's the type of person to find humor in the darkest of moments, while she would rather marinate in her gloom. If it comes to it, Gimli would flat-out reject whatever treasure offered to him in lieu of interacting with Cersei, which is saying quite a lot given the Dwarven penchant for treasure.
Would Work: Legolas & Brienne
Brienne and Legolas aren't very complementary people, which isn't strange considering their drastically different upbringings and experiences. She is rejected by her family, whereas he leaves home of his own volition.
At the same time, both these characters are given to long periods of quiet rumination, and there is little better than hanging out with someone who is also interested in nothing more than comfortable silence. There is scant chance for this friendship to turn into romance, but it's not entirely out of the question.
Would Turn Ugly: Sansa & Galadriel
Galadriel was once not the person she is in the trilogy — on the contrary, she used to be proud and self-centered during her youth in Valinor. She remembers that when she denied Fëanor her hair, he went on to create the Silmarils, and the world consequently turned apocalyptic when the gems were stolen by Morgoth.
Galadriel will never be her old self again, which is exactly what she will see reflected in Sansa Stark's domineering facade. Forget friends, they won't even be pen pals.
Would Work: Aragorn & Jon Snow
Jon Snow and Aragorn becoming fast friends is not only possible, but inevitable. They are both self-assured young men (well, 87 is quite young for the Elessar), as well as exhibit a sense of honor and dignity shared by few others in their respective worlds.
Of course, the word "honor" in Game of Thrones is often meaningless, but the point is that Jon is one who keeps his word regardless of anything. Aragorn may not have grown up in a hostile environment, but that doesn't mean that these two won't have anything to bond over.
Would Turn Ugly: Petyr Baelish & Gandalf
Petyr Baelish is one of the smartest people in Westeros, a feature that will place him at a similar intelligence ranking among LotR characters as well. Gandalf also holds a similar distinction, but one of the many ways in which they diverge from each other is in the spirit of compassion.
Littlefinger thinks of nothing other than himself, while Gandalf cares about everything except his own personal problems (if any.) The Grey-turned-White Wizard will absolutely not tolerate Petyr Baelish's cruel scheming and the latter would not enjoy being constantly babysat.
Would Work: Merry, Pippin, & Arya
Merry and Pippin don't exactly quantify as badass, like Arya does. However, assuming that their lives are stable (as at the conclusion of either story), the three of them are liable to go on their own wild escapades together.
While she has been hardened by suffering, Arya remains an adventurer at heart, explaining why she leaves Westeros to discover more about her mysterious world. She might as well take Merry and Pippin with her.
Would Turn Ugly: Joffrey & Eowyn
It's hard to imagine anyone becoming friends with Joffrey, considering that his nature can be classified as uncouth at best (and malevolent at worst). Maybe someone like Grima Wormtongue would take a shine to him, but not the noble, courageous Eowyn.
She spends so much of her life fighting a battle within a battle — to prove herself as a woman in a man's world, and one that's about to be destroyed if she doesn't fulfill her destiny. Joffrey, on the other hand, boasts about killing people but has never set foot in the vicinity of a battlefield. Eowyn would, and should, eat him for breakfast.