Renowned Selective Mentor

Occasionally The Hero will be mentored by someone who holds an important position in their community. It may be the group's leader, someone with a special job that only takes on one trainee in their lifetime, or perhaps an old hero who is retired and normally no longer trains anyone. Sometimes the mentor is chosen by someone else, but for the most part it is their own decision to take the character under their wing. Perhaps they see some sort of special ability in the hero. Perhaps, especially if the character is young, they want to be an influence in the pupil's life, trying to make them grow up in a certain way.

Whatever the reason, it is almost always a big honor to be chosen by this person. Because of this, the choice may result in someone becoming jealous of the hero, possibly turning them into a rival.

May have some overlap with The Obi-Wan. Mentor Occupational Hazard may also occur.

Examples of Renowned Selective Mentor include:

Anime and Manga

The Forwards in Nanoha Striker S have been handpicked for a year-long personal training by the eponymous character. Most combat mages in the setting consider themselves lucky to get Nanoha to act as their OPFOR for half a day. Not to mention that she normally only teaches advanced aerial combat to squadron-sized classes (as opposed to one-on-four training from basics up for the Forwards).
Jiraiya in Naruto fits this trope to a T. He only had two groups of students before Naruto, and in both cases one of them was an insanely gifted shinobi who became a legend. He chose to train Naruto become he reminded him of the Fourth Hokage, his previous student and Naruto's father.
- Kakashi can be considered this, as until Team 7 came along, no one passed his Genin Exam.
In Yu Yu Hakusho, Yusuke has to enter and win a tournament in order to train under Genkai. Unfortunately, she has no concern for the moral character of her student and successor, so Yusuke must also win to prevent the technique from falling into the wrong hands.
Koumyou Sanzo from Saiyuki puts his chosen successor, Kouryuu (later Genjyo Sanzo)) in a tough place by making him the favored disciple. The rest of the monks at the temple are explicitly jealous, and spread rumors of Koumyou choosing him for other reasons

Fan Works

In the Transformers fanfic Things We Don't Tell Humans, Terratron is this.
In the Buffy the Vampire Slayer Peggy Sue fic I Am What I Am, one of the flashbacks shows how Xander became the student of one such master, who knew about the supernatural, believed Xander when he said he'd been fighting vampires since he was a teen, and understood what he meant when he said he wanted to be Batman.

Film

In Dragonheart: A New Beginning, Geoff is trained by the king's advisor Osric. Osric claims it is because he's a natural and deserves special attention, but his real motive was to make Geoff eager for battle and to trust him in order to convince Drake to give up half his heart.
Star Wars: Due to his unusual affinity with the Force, Anakin gets mentoring and attention from high-ranking Jedi beyond that given to other padawans.
Pai Mei from Kill Bill. Apparently he only rarely accepts students and is a thousand-year-old renowned recluse.

Literature

In The Giver by Lois Lowry, the community only has one titular Giver at a time. Each Giver must choose a child as his successor during his lifetime. The main character Jonas becomes the Giver's student, and he is considered to have a special rank in the community.
In Warrior Cats, it is considered to be a huge honor to be mentored by the Clan leader or, to a lesser degree, the deputy. It occurs only a couple times in the series, most notably in the first book when Bluestar, the leader of ThunderClan, chooses the main character Firepaw as her apprentice. It is also considered an honor to train as the medicine cat's apprentice, because it is such an important position; each medicine cat only trains one apprentice in their lifetime. In that case, however, it usually isn't a surprise because the younger cat already has an interest in healing and helps out the medicine cat for a while before officially being apprenticed.
Miss Havisham trains Thursday for Jurisfiction in Lost in a Good Book. She's specifically described by Mrs. Dashwood as being highly selective, and she herself says as much, warning Thursday that she could easily lose the privilege of studying with her.
Harry Potter has a much closer relationship with the headmaster, Dumbledore, than is usual for a student, to the point of the Professor being almost a surrogate father. In the sixth book Dumbledore even gives him special lessons.
In Tamora Pierce's Circle of Magic series, Tris is trained by Niko, the super duper famous mage. Justified, in that the rules of magic dictate it.
- Specifically, any mage that discovers magic in someone is responsible for teaching them or finding a mage with the same kind of magic to take over instruction. Mages who can control lightning like Tris are both rare and tend to get themselves killed young so Niko is responsible.
In Thief of Time, everyone is shocked when Lobsang becomes the apprentice of Lu-Tzu The Sweeper. Even if they don't recognise The Sweeper in person, they are shocked by his rep, and because he never takes apprentices anymore. In fact Lobsang's apprenticeship is a punishment posting for both of them in different ways.
Darian certainly has this reaction when he finds out Firesong k'Vala has moved to k'Valdemar Vale specifically to train Darian.
In Belisarius Series an emperor hires one of the greatest warriors in India to train his daughter as a Badass Princess. Later she and her mentor get married and they become a Battle Couple.
- Toyed with later in the series. Rajiv was the eldest son of the near legendary warrior Rana Sanga, and wound up learning from one of the only two men who had ever even survived a duel with him... a lowborn Roman soldier named Valentinian.
In Trudi Canavan's The Black Magician Trilogy, the heroine, Sonea, becomes "The High Lord's favourite". He supposedly takes an interest in her unusually powerful magic and talent (and he does) but it's actually also so he can keep an eye on her and use her to blackmail her friends into keeping his ability to use Black Magic a secret.

Tabletop Games

In the Old World of Darkness RPGs, placing four or five dots into the "Mentor" Background will give one of these to your Player Character.
In Forgotten Realms the Chosen of Mystra usually have this status. Even if they don't want to - Mystra's main dogma is all about promoting magical knowledge and practice, so it's not like her most trusted servants are disinclined to teach. And it's not like no one volunteers - between being either selected or designed for natural talent, learning directly from the gods of magic and having superior capabilities given with a little speck of divine power they are high-end magic users. And whether others know who and what they are or not, witnessing one in action quickly creates reputation, then droves of questing wannabe apprentices coming with it. It's just that their job includes everything for which common priests are not enough and an avatar is not quite called for, or would be inconvenient for some reason (such as other gods paying too much attention). Which makes them very busy people (though having no need to sleep helps), and very unsafe to hang around. Those with well-defended bases (like the Witch Queen) keep a few apprentices at all times, the rest rarely take one. It doesn't help that shy apprentices rarely apply to the high and mighty, and bold ones "tend to glom onto things that are best left unglommed" (as Elminster commented on one candidate).

Western Animation

Master Piandao from Avatar: The Last Airbender, skilled swordsman, former Fire Nation Elite Soldier (but not a FireBender), High Master of the White Lotus Order and famous for defeating 100 soldiers alone after he deserted the army, there were many people who wanted to be trained by him, but he rejected most of them, between the few pupils he accepted to train were Prince Zuko, and Sokka (who gained him a level in Badassery after his training).
My Little Pony: Friendship Is Magic: Twilight Sparkle was just trying to get into the royal academy. One magic overload later, she is studying directly from Princess Celestia, the ruler of Equestria herself, who also happens to be something of a Physical Goddess.
- Note that this is as much a matter of raw power as skill—Twilight's Mega Manning would present a serious risk to those around her if she didn't learn proper self-control.

Real Life

Bruce Lee went through a period of this... due to having a surplus of people who wanted to learn from him.