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In Naruto, not only is Naruto the son of the Fourth Hokage, he's also a descendent of the legendary Sage of the Six Paths through his mother Kushina's clan, the Uzumaki (whose members inherit incredibly strong recuperative powers and an lengthy lifespan).
Other characters descended from the Sage through the Uzumaki are Karin and Pain/Nagato.
The Sage, through his two sons, is also the ancestor of the two founding clans of the Hidden Leaf Village; the Uchiha (whose members include Sasuke, Itachi, Madara, and Tobi/Obito, and the Senju (whose members include the First, Second, and Fifth Hokages).
Tsunade was partly chosen to be Fifth Hokage precisely because she was the granddaughter of the First. Early on, she's less than thrilled with the notion herself, as it just means she's from a lineage of people who got themselves killed in battle.
The Sage's twin brother also had descendents of his own; the Hyuuga clan (whose members include Hinata and Neji), and Toneri, the Big Bad of the canon tenth film.
Hellblazer - The Constantine family tree is an example. The lineage is known in history as Laughing Magicians; mortals who exist only to bluff, trick, and humiliate gods. This family existed just after the creation of gods and existence. Though not always seen as heroic, many of the family makes up a great Magnificent Bastard. Not to mention saving the world countless times.
"Inferior Five" - inverted, as the comedy comic is dealing with the misadventures of the children of the ersatz Justice League - who are comically inept
Made into an actual game mechanic in the Saga Edition tabletop game with the Legacy Destiny, where your normal Destinynote Something like Destruction, Creation, Redemption, Prevention, Sacrifice, etc is replaced with a family line that grants you a special ability (usually, getting a Critical Success more-or-less when you really need one).
The Lord of the Rings has a classical example with Aragorn, whose lineage is certifiably heroic going back about 70 generations, and originates from the three 'fathers' of both Elves and Men. On a minor note, Eowyn also argues that she's entitled to fight because of this trope.
Aragorn is also descended from Melian the Maia (an angelic being similar to a demigod) giving him Elvish, Human and Divine descent.
On the other hand, Aragorn's human lineage originates in the people who made Sauron their high priest and who were so evil the powers that be tried to Kill Them All by sinking half of Middle-Earth. His elvish lineage makes him a Noldor, who betrayed the Valar, murdered their family to steal their ships and either followed Feanor into evil or hid themselves away and did nothing to stop Morgoth's rampage.
So deeply-ingrained is this trope that, for years, legions of Harry Potter fans saturated the Internet with theories that Harry was the descendant of Godric Gryffindor ... this, despite how the series' prevailing Aesop is that people should never be judged by their ancestry or background.
It didn't help that his enemy Lord Voldemort had Villainous Lineage, being a descendant of Salazar Slytherin.
As it turns out, Harry is the last living descendant of Ignotus Peverell, the noblest and cleverest of the three Peverell brothers... which, frankly, makes Harry the closest thing the three deathly hallows have to a proper heir.
Historia Regum Britanniae chronicles what happens to a splinter group of survivors from the Trojan War, which eventually settles in Britain. The name of the hero is Brutus, whose name eventually became the name for the Britons.
In Dune, House Atreides claims lineage from King Agamemnon of Argos (or Mycenae, depending on the source), who fought in the Trojan War. This claim originally belongs to the Titan Agamemnon (not his real name) and taken up by his son Vorian Atreides, whose last name is derived from "Atreidae", the collective name for the descendants of King Atreus, Agamemnon's father (who was anything but heroic).
In Warrior Cats, almost all of the point of view characters in the main series are the descendents of the living legend Firestar, the first hero of the series.
Deltora Quest, all kings and queens of Deltora are descendants of King Adin, a blacksmith who united the Seven Tribes to drive back the Shadow Lord's army. Searching for the King Endon's child was a part of the main quest in the first series. It's Lief, Dain was actually a Grade 3 Olspy for the Shadow Lord.
All the Abhorsens are of the same bloodline, so both Sabriel and Lirael of the Old Kingdom series are descendants of heroes.
In Gosei Sentai Dairanger, Lin (the present-day Hou-Ouranger) is descended from the Dai tribe, and Ryou (the present-day Ryuranger) is of more direct lineage: He's the son of the original Ryuranger.
The present-day Kakurangers from Ninja Sentai Kakuranger are each descended from their respective predecessors, who sealed away the Youkai Army Corps long ago.
Samurai Sentai Shinkenger and Power Rangers Samurai feature samurai whose families have fought the monsters for generations; the Red Ranger from the Shiba clan and the others from the clan's retainers. When a friend of the Red Ranger wishes to join them as the Sixth Ranger, much is made of the fact that he is self-taught and has no such lineage.
Saban's Masked Rider, a Power Rangers spiritual spin-off, starred Prince Dex of the planet Edenoi, the latest in a long line of Masked Riders, with the powers passed down the royal family for generations. The audience is introduced to his ten predecessors in "The Invasion of Leawood".
Legend of the Five Rings RPG, famous ancestors can be bought as merits at character creation. Each of them gives some merits (and sometimes drawbacks) depending of what he was famous for.
Lords from Fire Emblem as a whole universally qualify. The 4th game makes this an actual game mechanic: characters who are descendants of one of the Twelve Crusaders or the Dark God Loptous receive boosts to their growth rates, weapon levels, and can even use their respective ancestor's Ancestral Weapon in certain conditions.
Susano from Ōkami is a descendant of the legendary hero Nagi. His resentment and later acceptance of his ancestor has a major impact on the plot of the first half of the game.
Link and Zelda from The Legend of Zelda series are usually not the same exact people in each game, but different incarnations from different time periods.
All of the Zeldas are blood descendants of the Zelda from Skyward Sword, who was a reincarnation of the Goddess Hylia.
The Links are not necessarily blood-related but are said to share the "Spirit of the Hero" and are at least a spiritual lineage of heroes. Only the Links from A Link to the Past and Twilight Princess are explicitly stated to carry a special bloodline, and according to the Hyrule Historia guidebook, the only confirmed relation between Links is that Link from Twilight Princess is a descendant of the Hero of Time from Ocarina of Time.
The Wind Waker gets special mention because it deliberately subverted the trope, revealing that its Link was not related to the Hero of Time's bloodline at all. Many of the powers that be doubt that he can save the day for precisely this reason, and Link proves them wrong.
Dragon Quest I had you play as the descendant of the legendary hero Loto. Dragon Quest II had you play as the descendants of the first game's hero. Dragon Quest III had you play as Loto, who was himself the descendant of another hero. Somewhat subverted in Dragon Quest V, where the legendary hero is actually a descendant of the main character, who marries a descendant of the last legendary hero.
Shiki Tohno in Tsukihime is a last descendant of Nanaya household, a lineage of pure-blooded humans (read: "weakest species in the setting"), who trained their bodies and honed their skills enough to fight most supernatural threats, be it Demons, vampires, or what have you.
In Fable it is implied that all heroes are descended from the first Archon.
Which leads to a Crowning Moment of Heartwarmingand Tearjerker when the supposed ancestor himself is executed, and with his last words acknowledges the heroes as the true "descendants" of his heroic spirit.
Ico is the descendant of Wander, protagonist of Shadow of the Colossus. Though Wander's fame is debatable, and the one sign of the lineage (the horns) is more of a curse than a blessing for Ico.
Rarely mentioned, but all the humans of Gensokyo (that didn't stumble from The Outside, anyway) are descendants of brave people who defeated the youkai to settle there.
The Belmont clan from Castlevania, always out to hunt the night. Most of the series' protagonists so far have been Belmonts. The Belnades/Fernandez clan get to do their share of vampire-butt-kicking over the years as well.
In Dragon Age II, the Legacy DLC reveals that the protagonist's deceased father, Malcolm Hawke, was once tasked with fortifying the seals of an Ancient Grey Warden fortress. Because this required his blood in order to do so, this meant that any member of his lineage became keyed as the only ones capable of wielding "The Hawke's Key"; making them capable of either reinforcing or unlocking the seals containing Corypheus.
Jak and Daxter: Jak's father is "the great warrior" Damas. Through him, he is a descendant of the historical founder of Haven City, Mar.
Sir Anduin Lothar, one of the greatest human heroes in the Warcraft franchise, was the last descendant of the Arathi royal bloodline which founded the first human empire on Azeroth. Sir Lothar took advantage of this to convince the High Elves to join the Alliance since the High Elves were indebted to the Arathi.
This is a gameplay mechanic in Massive Chalice: the demon-fighting heroes and heroines are supposed to retire at some point, settle down and get children who will inherit some of their heroic abilities and continue the fight. Since each new generation is slightly stronger than their parents, this is required to check the ever-increasing demon threat.
In Ni no Kuni, Esther is the daughter of the great sage Rashaad, a legendary mage known for his many heroic deeds. Actually, every playable character is the son or daughter of a Great Sage: Oliver is Alicia's son, Esther is Rashaad's daughter, and Swaine is the son of the mage-emperor of Hamelin. In the PS3 version, you eventually also get Swaine's brother Marcassin, who shares the same lineage.
Played for laughs in Cucumber Quest: Cucumber's father Cabbage tells his son that the men in their family have been heroes for generations, in order to convince him to become a hero as well. When Cucumber points out that Cabbage isn't a hero, he says that "it skips a generation here and there." He's not wrong though.The world of Cucumber Quest is stuck in a Vicious Cycle where the Nightmare Knight gets resurrected every 5,000 years, and every time a hero (one of Cucumber's ancestors) comes to stop him. The story takes place during his 100th resurrection.
Zig-zagged in Girl Genius, where our heroine Agatha Heterodyne's father and uncle were indeed Legendary Heroes... and the White Sheep of an unbroken line of vicious warlords that were the terror of Eastern Europe for centuries. And then there's her mother'sside of the family....