Heir to the Dojo

A convenient excuse for the Ordinary High School Student to have high-powered martial arts skills is for the student to have been training since a very young age, and to be the heir to some obscure martial arts school. The current master may be a parent, or just a close friend who has taken the student in as a disciple. Unlike school clubs, martial arts training takes place outside of school, so the Heir to the Dojo is unlikely to be a club member.

The dojo itself is generally part of a Big Fancy House compound that features The Thing That Goes Doink. Challengers frequently arrive to test the skills of the dojo's style... and the Heir to the Dojo is generally put forward as the school's champion.

A weaker form occasionally crops up in more Western works: an unlikely character ends up wiping out someone or two, and explains that their *insert relative here* is a sensei/boxing trainer/wrestling coach/gunslinger, and he/she's always grown up learning karate/boxing/wrestling/marksmanship.

Examples of Heir to the Dojo include:

Anime and Manga

Kenshiro of Fist of the North Star is probably one of the oldest examples. He is the legitimate heir of Hokuto Shinken, the most powerful martial art seen in the series, and regularly challenges the successors of others martial arts styles (mainly heirs to the different branches of Nanto Seiken, the 'rival' school of Hokuto Shinken).
Ranma ½ provides several examples. Not only are Ranma and Akane the heirs to their families' particular branches of Musabetsu kakuto (Anything-Goes Martial Arts), they frequently encounter the heirs of other, often bizarre or outrageous, styles. It's only natural, however, that practitioners and aficionados of bizarre martial arts would seek each other out.
Both Nagisa and Nosaka from Chou Kuse ni Narisou
Jubei-chan
Kobayashi from Tenshi na Konamaiki
Kamiya Kaoru in Rurouni Kenshin
Koushi and Momoko are both heirs to their respective dojos in Sumomomo Momomo, though Momoko's entire motivation revolves around not feeling (and not being considered by her father) to be worthy of the dojo.
Initial D, but instead of martial arts, racing street-legal cars on mountain roads is the art involved.
Yawara might provide the world's best example of this trope.
In addition to being a Magical Girl, Nanoha of Magical Girl Lyrical Nanoha also comes from a family with its own martial arts style and dojo; her two elder siblings are quite proficient at the style.
- And Kyouya's (and Miyuki's) mastery of the Mikami-Fuwa-ryuu style figures prominently into Triangle Heart 3 ～sweet songs forever～.
Aoyama Motoko in Love Hina, after her sister Tsuruko decides to get married instead of inheriting the family dojo.
Saeko Busujima in Highschool of the Dead.
Ninja-girl Kaede Nagase in Mahou Sensei Negima. According to her character design notes, it was originally suggested that the rest of her family gave up on the art long ago, but in the manga, she refers to herself as having the rank of Chunin.
Honey from Ouran High School Host Club is heir to a large martial arts family in whose specialty he is extremely proficient.
Subverted in Naruto in that a large part of Neji Hyuga's Arrogant Kung Fu Guy persona stems from the fact that he is not Heir to the Dojo but a curse-marked member of a branch line.
- Played straight with Sasuke early on; half the crowd at the Chuunin Exam seemed to be there to see "the Uchiha" show his moves. Then again, pretty much his whole clan was wiped away. Except for Itachi, but that was another huge bag of cats...
Chiko of The Daughter of Twenty Faces is heir to a "dojo" of sorts, if you can call an international thieving ring a dojo.
Subverted in The Prince of Tennis: Rikkai sub-captain and star tennis player Sanada Genichirou is actually heir of a famous kendo dojo, but tennis is his main activity. Same goes to martial artist Wakashi Hiyoshi from the Hyoutei team.
Subverted in Nanaka 6/17, where Jinpachi Arashiyama is the actual heir to the dojo as the oldest child, he doesn't know the first thing about kendo. Meanwhile, his sister Satsuki has many of the characteristics commonly associated with this trope, including being a super-intense Arrogant Kung Fu Guy dedicated to becoming the best. She still encourages her brother, a tough-as-nails delinquent, to be as manly as possible to keep the family's reputation strong.
Masaki Tenchi seems to have this as a reason for his skills with a sword, although his teacher hangs out at a shrine instead of a dojo. This doesn't explain other issues, just the thing with the sword.
- Makes more or less sense, since his teacher and grandfather used to be a mighty swordsman before he came to Earth and became the owner and priest of the shrine.
While Azumi Kiribayashi from Real Bout High School is the heir to her family's dojo, it's pretty far from her motivation as a fighter, in a bit of a subversion.
Yu-Gi-Oh! GX has an interesting take on this; Ryo Marufuji, outside of being the best duelist out of Duel Academy, is also the heir to the "Cyber Legacy", essentially the Tyke Bomb of the "Cyber-Style" of dueling that incorporates his legendary Cyber Dragon cards (and even has its own dojo in the mountains for teaching duelists how to use the style). This becomes a plot point in two ways: first, when he went back to the dojo and faced off with his former teacher (Duel Academy's principal) for the dark half of the Legacy (the Cyberdark monsters he uses as Hell Kaiser), and in Season 4, when the heir to a rival "Psycho-Style" dueling school challenges him to determine which dueling style will remain in practice.
Tyson/Takao of Beyblade
Tamaki Kawazoe of Bamboo Blade is the heir to her father's kendo dojo.
Ken Wakashimazu from Captain Tsubasa is heir to a karate dojo, and a good part of his early Character Development has him tyring to not be heir anymore. He loves karate, yes, but loves soccer even more, and his dad gives him a year of probatory to prove this to him.
Gangryong from Veritas is heir to the Enlightenment Of Thunder and Lightning (EOTL) school of Korean martial arts. This is subverted a bit that Gangryong didn't know he was the heir to this until he was told that his master was killed.
- A large portion of the student body is this, being the children of dojos that follow the rule of "only one successor" and all that.
When Fumio's grandfather in Saitama Chainsaw Shoujo handed down his Chainsaw Martial Art to his granddaughter, it was presumably to make things easier when she took over the family lumber business rather than for slicing up her classmates.
In Gintama, while subverted, Shinpachi and Otae inherited their dojo after their father's death however they are very poor. And nobody really studies there, either. Kyuubei and Kondo, on the other hand, play this trope far straighter.
In Kenichi: The Mightiest Disciple, there is a Fanon theory that Kenichi one day might become a heir to the Badass Grandpa Hayato Furinji as the head of the Ryozanpaku dojo. This trope is, however, played (almost) straight in the case of the Evil Mentor organization Yami: some disciples of Yami masters are candidates to become the heir to the Big Bad, the problem is that the masters cannot agree which one is the best candidate.
- Freya is the inheritor of a particularly obscure school of staff-fighting.
Myoudouin Itsuki was made into this since her sick older brother isn't expected to make it. She certainly has the skills, but in order to become as respected as he was, she must hide her femininity and love of cute things beneath a masculine exterior...and it's clearly wearing on her.
- Thankfully, though, her brother does make it and the first thing she does is become a Third Ranger... er, Cure.
Jin from Samurai Champloo used to be one of these.
Miko Mido in La Blue Girl. The kunoichi of the Mido clan not only learn ninjutsu, but sexual spells. Miko is the heir to the Mido clan, and has been training. (Even though, especially in the anime/OVA adaptation, she'd rather just be a normal teenage girl leading a normal life.)
Subveted in Soul Eater, where there is the Higly Visible Ninja Black Star, last heir of the Boshi Clan, which was destroyed when he was a newborn, so he didn't learn anything from it, but is anyway very powerful
- And played straight by Black Star's weapon, Tsubaki, who is direct descendant of the first Demon Weapon.
Makoto from W Juliet has a problem that's similar to the one faced by the aforementiioned Ken Wakashimazu: he does not want to be this despite his great skills, and his father is putting pressure on him. Makoto also gets a chance to get out of the tradition, but in his case, his dad tells him that since looks very girlish and he wants to be an actor, he shall prove himself as skilled enough to spend his highschool years dressed as a girl without having his identity revealed to the public. If his true gender is leaked out, he'll come back home and retake his Heir to the Dojo place without complains.

Comic Books

In the "Marvel Mangaverse" Alternate Universe - a series of comics with Marvel characters re-imagined manga-style - Peter Parker is secretly the last ninja of the Spider Clan.
Ichikun Ichinohei (a.k.a Itchy Koo) from Ninja High School

Live-Action TV

Played with in Power Rangers Ninja Storm. Cam, despite being the son of the sensei of the Wind Ninja Academy, was not given any ninja training due to a promise given to his late mother. However, he could still defend himself against the enemy Mooks, because "I grew up in a ninja school; do the math." Later on, the promise is worked around by training him as a samurai instead (even receiving the mother's blessing via Timey-Wimey Ball).
In Tokusou Sentai Dekaranger, we learn that Anubis "Doggie" Kruger was given the inheritance to the intergalactic dojo he trained at as a cadet over the dojo owner's own son, who was considered to be too out of control and prone to abusing whatever power he was given to be trusted with it. Not surprisingly, this proved to be a problem for Doggie, in a latter episode.

Video Games

Sort of occurs in Pokémon Ruby and Sapphire. The protagonist's father is the leader of one of the gyms.
- While this example carries over to the rest of the franchise even less - both May (from the anime) and Ruby (from the mangas) become coordinators - two filler episodes of the anime featured dojos based this trope.
- More traditionally, Janine is heir to the gym in Fuchsia City, and takes over when her father Koga becomes a member of the Kanto/Johto Elite Four. The anime version of Flannery also recently inherited the Lavaridge Gym from her grandfather (in the game and manga, her grandfather was a member of Hoenn Elite Four, but may or may not have been the Lavaridge Gym Leader).
- Another Pokemon example. The Opelucid gym in Pokémon Black and White splits this across two games, but the two leaders have the same team. In Black, you face Drayden the master; in White, the apprentice Iris is your opponent.
Ryo Hazuki from Shenmue.
Street Fighter III: Makoto.
While it looks like your character (and Dawn Star) in Jade Empire are this, the truth is much, much worse...
Ryo Sakazaki of Art of Fighting fame seeks to inherit the title of "Master of Kyokugen" from his father, Takuma. In some continuities, he succeeds (such as when he is portrayed as a much older "Mr. Karate" in Buriki One) and in others he prepares other disciples to do this (such as Marco in Garou: Mark of the Wolves.)
Anna Lemouri of Mana Khemia is heir to her family dojo. In her final character subquest, she is trying to decide whether to dedicate her life to alchemy or swordsmanship. In her ending, she integrates alchemy into her school of swordsmanship - which means that she uses explosives and poisons in addition to blades.