Having Mixed Ancestry isn't fun. It's rather lonely not fully belonging to the societies of either parent, even if there isn't active discrimination against what you are. However, if one or both sides of the family have some special power or supernatural trait, there may be a bonus to being a half-breed: a powerball in the Superpower Lottery.
Blame Powers as Programs or LEGO Genetics, but any power possessed, even latently, by a parent, will always be passed on to their offspring, and the presence of a different strain on the other side of the family will only serve to amplify it. This is especially true if the powers are complimentary in some way, but even if they are opposite: if someone with fire powers marries An Ice Person, their kid may well have both sets or even an amalgamated access to Pure Energy, a living Yin-Yang Bomb.
You might expect this to be less the case with Half-Human Hybrids or other combinations of the supernormal and the mundane, but this is not always so. Often as not, the mundane side will leave all of the powers intact, but mitigate any weaknesses, quite likely resulting in One Hero, Hold the Weaksauce. The Dhampyr is perhaps the ultimate example of this.
It's usually the Rule of Cool in action, but also Truth in Television to some extent. Mixing genes that don't normally mix can have unpredictable results. A good example is the liger — mating a male lion to a female tiger produces offspring that grow to be far larger than either parent.
Other than reproduction, this kind of advantageous combination can result from a Fusion Dance. A Hybrid Monster might be bred specifically to take advantage of this trope. Contrast All Genes Are Codominant, in which special traits are exactly half-inherited by the offspring. See also Lamarck Was Right.
See The Other Wiki's pages on hybrid vigor and Heterozygous advantage for some information on the real-life equivalent.
Ichigo is the offspring of a human and a shinigami. Thanks to the quirks of his parentage, he therefore possesses all the known power types of the story. Ginjou does claim Ichigo is not the first human/shinigami offspring to exist. His mother was a Quincy whose soul was affected forever by one of Aizen's hollow experiments, and to save her life, his father underwent a procedure to change him from a pure shinigami into a shinigami/human hybrid. As such, Ichigo was born with quincy, hollow, human (fullbringer) and shinigami power, all of which have awakened at different speeds throughout the story.
When a person survives a hollow attack, they are left with a residual taint of the hollow's powers in their body. This is then passed on to an unborn child via the mother. The child is permanently affected by this, resulting in the birth of a fullbringer: a human being with hollow-originated powers and the ability to manipulate the souls of matter (whether living or inanimate) to achieve all sorts of power effects and outcomes.
This is also the premise behind Visoreds and Arrancar. The former are Shinigami who have acquired Hollow powers through artificial means, while the latters are Hollows who have broken their Masks in order to regain Human forms and Shinigami-like power.
Saiyan-human hybrids have far more inherent power than pure members of either race. The effect seems to get stronger with successive generations. At age five, Gohan could flatten mountains and stagger Freeza if he got angry enough. Goten and Trunks could go Super Saiyan without even thinking about it.
This is also the purpose of the Fusion Dance, the resultant character being a compression of the power of the original two, having Power Levels several orders of magnitude higher.
InuYasha: The eponynous half-demon is more powerful for it. Due to his incredibly powerful father he has more power than most demons, and his human side grants him use of the sword forged from his father's fang, which lets him take out any demons his own powers can't take.
This trope is the entire basis of Rikuo and Rihan's ability, Matoi in Nurarihyon No Mago. Being part human they can be possessed by their youkai subordinates but because they're also part youkai they have the ability to control the "fear" aura of the one possessing them, resulting in a Fusion Dance type situation.
Yusuke of YuYu Hakusho is revealed to be the descendant of a very powerful demon who had fallen in love with a human. After his demon genes become active, he gains the ability to use both Demon Energy and his original human-based Spirit Energy, eventually creating something akin to Sensui's Sacred Energy by combining the two.
Blade, a classic dhampyr, has all the strengths of vampires, but none of the weaknesses, barring a much weaker desire for blood which he treats with a serum.
Magnus, an alternate son of Magneto and Rogue, has the power of magnetism from his father, and a variation of his mother's deadly touch; in this case skin to skin contact will cause the victim to turn to pure metal.
The Witchblade is a living weapon that has been used by females, and some males, over the generations. It was the offspring of the Darkness and Angelus. It has the power to create a blade, fire energy beams, create wings for flight, and can cover the host in body armor, most of the time. It also doesn't have any of its parents' weaknesses; The Darkness can't function in light, while the Angelus can't fight in the dark.
Little Nicky, being the offspring of Satan and an angel, can perform both good and evil miracles.
In Sky High, the protagonist thinks he's powerless because his abilities are rather late to bloom. Turns out he gets both his dad's brute strength and his mom's flight (at different times), making him a Flying Brick.
The overarching story arc of the Underworld series is pretty much driven by this trope, though only the fourth movie features a child inheriting both parents' powers—other hybridization instances are carried out via vampiric blood absorption/transmission. To boot:
Michael Corwin becomes a werewolf-vampire hybrid by getting subsequently bitten by Lucius, a werewolf, and Selene, a vampire.
Selene becomes a vampire-Immortal hybrid (among other things, rendering her immune to sunlight) by drinking Alexander Corvinus' blood in the second movie.
The fourth movie reveals that Selene and Michael's daughter inherited both her dad's werewolf strength and her mom's vampiric abilities. Since most of Selene's pregnancy occurred after she became an immortal, it is likely that Eve inherited immortality, as well, making her a triple hybrid.
Demigods in Percy Jackson and the Olympians are often chosen by the gods to do their work because being half-god and half-mortal, they straddle both worlds and can go places and do things that either mortals or gods cannot.
The Silmarillion: The offspring of Beren [a Man] and Lúthien [an Elf with Maiar heredity] have the ability to choose which race they belong to, which determines their ultimate fate. Even if they choose the human side, they can expect to live hundreds of years.
On Charmed, a common witch power is telekinesis and a common power for Whitelighters is orbing. Paige, Chris, and Wyatt, who are all half-witch/half-whitelighter, they each combine these powers into orb-telekinesis.
In the fourth series finale of the rebooted Doctor Who, the Doctor undergoes a "biological meta-crisis" in which he uses regeneration energy to form a clone which has some of Donna's personality. More importantly, the same process gives Donna the Doctor's mental prowess, which, combined with her "gut instinct" as a human, makes her a hypergenius capable of things even the Doctor would never think to do. Unfortunately, the human-Time Lord meta-crisis proves to be unstable, and would kill her if she didn't have her memory erased.
Inverted in Farscape by Scorpius, the offspring of a Sebacean and a Scarran. He inherited both the Scarran bodily trait of massive internal heat generation, and the Sebacean inability to regulate body temperature easily—requiring him to use cooling rods inserted in his head to survive, which created a certain vulnerability. Other than that, he was quite Badass, however, both physically and mentally.
His hybrid physiology does allow him to distinguish heat signatures, however, which allows him to distinguish species and tell when someone is lying.
Supernatural: Jane the Nephilim was physically stronger then Castiel or Metatron and tosses them around in the fight.
Connor in Angel is the Dhampyr version...he has the strength and enhanced senses of the vampire with none of the drawbacks.
In Munchkin, this happens whenever a player uses Half-Breed with a single race card — he gets bonuses against anyone that race card gives bonuses against, but doesn't take penalties when fighting monsters who have an advantage over that race.
Devil May Cry 4: Angus wonders why Dante was more powerful than he. While he [Dante] never gives Angus a straight, it pretty munch boils down that he [Dante] has the power of the demons and determination or soul of a human that makes him powerful.
Dm C Devil May Cry has a better example with that version of Dante who is half demon and half angel, and thus has access to weapons with both demonic and angelic power.
In Diablo, all humans on the world of Sanctuary are the descendants of the Nephalem, offspring of renegade angels and demons. The Nephalem had power that potentially far outstrips their parent races, and in fear of them the angels tuned the Worldstone to diminish their power, resulting in the mostly ordinary human race.
Fire Emblem Awakening provides players with the ability to pair units in marriage, and allows their offspring to inherit their last equipped skill (one from the father and one from their mother) as well as letting their base stats be influenced by the stats of the respective parents at the time the player encounters each kid on the game.
In Fire Emblem Tellius, the offspring of Beorc and Laguz are known as the Branded (from the marks they have somewhere on their bodies, usually kept hidden) or Parentless. They often have special abilities, but it's generally considered a great stigma, until Radiant Dawn when it turns out the Apostle and Empress of the Begnion Empire is a Branded, the descendant of the Laguz Heron Lehran and the Beorc woman Altina, which enables them to hear the voice of the Goddess Yune. Also, aside from Ike's army, most of those unaffected by Ashera turning the entire population to stone are Branded due to her having no idea that they exist.
A variant in Pokémon games via Hot Skitty-on-Wailord Action: Breeding two mons together produces a mon of the females' species, that will learn some of the male's moves. So in theory the new mon might have some more powerful moves (sometimes at level 5, meaning just-hatched) without any drawbacks except that hatching it takes longer.
In StarCraft, two alien races were uplifted by the Xel'Naga: The protoss, who with their powerful bodies and Psychic Powers, are deemed to have "purity of form", and the Zerg, which with their mutative adaptibility, Hive Mind and resultant singularity of purpose, have "purity of essence". Eventually agents of a renegade Xel'Naga create a hybrid species, which have the strengths of each, supposedly making them the Ultimate Life Form.
Xenoblade: Half-Homs High Entia, despite their usually smaller head wings and lack of Pointy Ears, are just as long-lived and capable of manipulating ether as their pure-blooded cousins. What really sets them apart is that they are cured of the genetic "curse" that can eventually turn pure High Entia into mindless Telethia.
In Skylanders, Sunburn is a dragon-phoenix, which grants him rare powers like teleportation. This actually turns out to be a mixed blessing, as his abilities make him a target for unscrupulous wizards; part of the reason he joined the Skylanders is for protection.
In Shin Megami Tensei III: Nocturne, when visiting the Labyrinth of Amala, a Loa comments demons sacrificed their ability to change and grow in exchange for a form of immortality, while humanity kept it at the cost of remaining mortal. He passingly wonders what this means for the Player Character, the Demi-Fiend, who as a perfect fusion of human and demon, has the best attributes of both...
In El Goonish Shive, Uryuoms frequently have several parents adding their genes to an egg, as they don't have typical genders. This ultimately results in a character being a hybrid of four different creatures, conserving many of their powers.
Inverted in an episode of the original series where the Omnitrix is damaged, causing Ben to assume hybrids of his normal forms. In all cases, these new forms are worse than his normal ones.
Combining Four Arms (a humanoid with Super Strength and four arms) with Stinkfly (a giant insect with wings) resulted in "Stink Arms", a physically weaker version of Four Arms who was too heavy to fly.
Played with when Kevin is transformed into a Shapeshifter Mashup of all of Ben's forms. While none of his powers are as strong as the originals, Ben points out that he can combine them to use strategies that none of his forms could use by themselves (the simplest being that weak versions of Super Strength, Super Speed and Super Toughness equals "one mean punch").
According to an anecdote told variously about George Bernard Shaw and Isadora Duncan, Shaw and Sandra Bernhardt, Albert Einstein and Marilyn Monroe, etc., the woman says to the man, "You are the greatest brain in the world and I have the most beautiful body, so we ought to produce the most perfect child." He responds, "What if the child inherits my beauty and your brains?"
Ligers,The offspring of a male lion and female tiger. They are massive due to a defective growth inhibitor (up to 900lbs, compared to a 400lb tiger)and much more powerful than either parent. That said, such size might work against them were they ever wild. Interestingly, it doesn't work the same way with tigons, where the parents are reversed.