Game of Thrones featured a colorful and unique set of characters from different backgrounds, who were thrust together to deal with two major threats. As a result, some of them were quite evil, while others represented the best of humanity's goodness and benevolence. In the Star Wars universe, this would have made them prime candidates for Jedi training.
If any of these characters were Force-sensitive, the Jedi Order would have had exceptional members to add to their ranks. Some may never have risen to the rank of Jedi Master, but they would have been invaluable to the Jedi cause. Others might have eventually landed a spot on the Jedi Council, due to their wisdom.
Tommen Baratheon
Tommen has all the makings of an excellent Jedi, despite his young age and inexperience. If taken by the Jedi Order at a young age, his natural goodness and compassion would have made for an excellent Padawan learner, especially under the right Master's tutelage.
His raw Force strength would be another matter, but Tommen could definitely excel as a Jedi Consular, with strong diplomatic skills. His inability to be ruthless would act as a ward against temptation by the dark side. HIs only weakness was his inability to deal with traumatic loss, which is why he took his own life in Game of Thrones.
Davos Seaworth
Though Davos has a checkered past, he could have made for an excellent Jedi in the same way that Kyle Katarn did in the Expanded Universe material. Davos is no stranger to being around the seedier elements of society, but his devotion to a cause is heroic, nonetheless.
He also possesses a rare and genuine humility lacking in many other Game of Thrones characters. He recognizes his limitations, and is not afraid to voice them openly, in front of others. This tempered mixture of character traits would have equaled a Jedi with strong tactical skills. One of many things that makes no sense about Davos Seaworth is his lack of fighting skills, however, so it's a tossup as to whether he'd be rubbish with a lightsaber, or gain some amount of skill.
Brienne Of Tarth
As far as Jedi Knights go, they could do with Brienne of Tarth among their ranks. This powerful warrior is not only gifted in skill with a blade, but she's also used to dealing with adversity, and remaining true to her own sense of honor. She's already gone through the gauntlet, emerged tougher, yet never lost her compassion.
Yet, her insecurity could be her undoing, and Brienne made a lot of mistakes in Game of Thrones, which could have cost her everything. That's why she'd need some solid training by an accomplished Jedi Master capable of letting her see the bigger picture. Otherwise, her self doubt could lead to fear, and then to anger. What follows next could be catastrophic.
Barristan Selmy
As a loyal Knight in service to the kingdom, Barristan knew the value of honor, commitment and integrity. However, he also knew when the King he was fighting for had lost his merit, and become corrupt. Barristan got out while he was still able, and managed to evade Joffrey Baratheon's wrath.
This was proof that Barristan would have stayed true to his principles, instead of simply going along with the status quo. Joffrey was the Game of Thrones equivalent of Emperor Palpatine in many ways, and rather than allow himself to be corrupted, Sir Barristan rebelled.
Benjen Stark
Benjen Stark knew the value of commitment to a good cause, and he gave up his life in service of the Night's Watch to see it happen. That included abandoning all royal claims he had on the North, including the Lord of Winterfell title, which was considered highly prestigious.
Even when things went sour for Benjen, he stayed true to his mission to battle evil, in whatever form it took. A man of royal esteem who was capable of such self-sacrifice would have made for a truly worthy Jedi, as the Order frowned on personal ties and commitments of any kind.
Gendry
Gendry was nothing special in his own mind, despite his status as the rightful heir to the throne in his father's absence. Yet, he was content to live out his days as a simple man who went to bat for his country. A capable warrior, Gendry tempered it with a strong sense of charisma and humility, which would have made him a great leader.
As a Padawan, it's hard to imagine Gendry giving his Master too much trouble. In fact, he would probably have risen quite fast within the Order, though it's doubtful he would have been asked to sit on the Council. He was best suited out in the field, carrying out missions that would have guaranteed galactic security.
Jorah Mormont
Another character with a highly checkered past, Jorah Mormont made many, many mistakes during his time on the show, then pursued redemption in the hopes that he would win favor with Daenerys. The trials and tribulations he went through to get there were astronomical, including facing down a fate worse than death.
Yet, he emerged just as strong, and committed to his cause. However, as a Jedi, Jorah would have to learn how to abandon the idea of romantic ties. His emotions would easily get the best of him, which means he'd have to learn to let go of everything he held dear. If he managed it, he would have made for one seasoned Jedi.
Bran Stark
Bran Stark was always a good lad, but his encounter with the Three-Eyed Raven left him permanently changed. He became a vessel for the Raven's power, and as such, was not quite the same boy that he once was. He also became far less emotional, and focused more on facts, interpretation of visions, and a pragmatic solution to problems.
This would have made him an ideal Jedi Master, even if his relative lack of charisma might have ruffled some feathers inside the Order. His ability to interpret visions and see through mediums would have been amplified through the Force, making him a powerful clairvoyant capable of shaping galactic events.
Eddard Stark
Eddard had his personal demons, to be sure, but he was also excellent Jedi material. He did cut corners throughout his life, but that changed in later years when he became determined to live honorably, and hold people accountable for their actions. He also realized the burden of compromise in such a high ranking position.
He was also a gifted swordsman, making him a cunning warrior. As an all purpose Jedi, Eddard would have most likely been sent on missions designed to quell tensions, and keep others in line. He was firm, but fair, as evidenced by the many iconic quotes Stark made throughout his time on the show.
Jon Snow
Of all the would-be Jedi characters from Game of Thrones, Jon Snow would be the most ideal, and the most dangerous, at the same time. His lack of self-confidence in the show was gradually put to rest as more responsibilities were thrust on him. In many ways, his ascension through the ranks was not desired, making him a prime candidate in reality.
He is selfish in battle though, so he'd need to learn to reel in his emotions, especially if he feared losing loved ones. If he managed it, he would definitely sit on the Jedi Council within a short space of time. His natural leadership qualities were second only to his sense of compassion, commitment to duty, and of course, the protection of those not capable of defending themselves. That's a Jedi in the making, if ever there was one.