When you have a Crisis Crossover or a Massive Multiplayer Crossover, you'll need something in order to motivate the heroes into action. While having the Big Bads of the series teamup can work, sometimes you want an even Bigger Bad. One that requires a such a massive team up of heroism.
And then, why stop there? Why not go the full way and make a new Hero for this crossover? That way you can have a central conflict for the entire story. Plus, since you'll have a character tied to this story instead of the differing canons of the other series, you'll be able to have more character development for the main character all in all.
The Bigger Bad variety is somewhat more common than the new Hero variety. Crossovers with new Heroes, however, have a tendency to become a series in their own right, a Spin-Off that just happens to have crossover origins.
Oddly enough, rare in Fan Fic, likely due to the fear of them becoming Mary Sues — but see Original Character.
The Yes! Precure 5 movie and Heartcatch Precure movie both had original Cures, but the ones from Yes! were evil and the one in Heartcatch was just to set up a bit of backstory.
A well known Bigger Bad example is the Anti-Monitor from DC's Crisis on Infinite Earths. There were some new heroes as well—Pariah, Harbinger, the Monitor, Alex Luthor, Superboy-Prime, Lady Quark, and Doctor Light—but, with the exception of Doctor Light, they faded quickly after the event ended, only brought up when the writer wished to remind the readers of Crisis.
Lampshaded in Marvel's anthology Wha... huh?'s Identity Crisis parody - Identity Girl says she was created just because of that one.
Neron of DC's Underworld Unleashed. And the new hero is — Trickster. Yes, the reform of a longstanding villain. On one hand, outwitting a devil is probably the best stunt he could have pulled of; on the other hand, he doesn't want to go somewhere warm after he dies, after pulling that off. And Neron, despite being an obvious plot device to beef up lack-lustre villains, has since made several appearances.
The lead Rider of Kamen Rider Decade is this. He's the original character that serves to cross over elements of the nine series before him, partially by being able to make use of all their powers. His primary enemy, Narutaki, brings random enemies from the crossover elements to combat him. Diend, the second original Rider, serves as a rival who is less concerned with fighting Decade and more with stealing and trolling, but has the ability to send clones of crossover heroes or villains into battle for him. The major enemy group Dai-Shocker later appears, which is basically one or two people who got the smart idea to have all previous enemy groups in the Kamen Rider series stop fooling around and join forces.
The Grand FinaleMovie is a bit more open about this: Wataru Kurenai, the original universe Kamen Rider Kiva, tells the cast that Decade exists solely to be whatever the Rider worlds need him to be: hero or villain, savior or destroyer. He has no home and no story of his own, and therefore after he presses the Reset Button through Suicide by Cop, there's supposedly no way to bring him back.
Another toku crossover example is Go Go Sentai Boukenger vs. Super Sentai's AkaRed. He carries with him the powers of all the previous Red Rangers, and demonstrates as such by turning into GaoRed and MagiRed, as well as summoning weapons from other heroes.
All six rangers in Kaizoku Sentai Gokaiger can become any other Super Sentai hero. An enemy named Basco can summon them similar to how Diend did.
The Original Generation games decidedly goes recursive, as a number of original characters were created just for these games.
And now the Super Robot Wars Original Generation characters(At least Ryusei, Kyouske, and Masaki) have shown up in Another Century's Episode R, where they aren't treated as originals. Banpresto has found a way to make it go full circle.
In Kingdom Hearts, you have Sora and his friends to make the Heroes. There are a few other side characters, and then the Bigger Bads: the various incarnations of Xehanort, Organization XIII, and of course in general, The Heartless.
The King Of Fighters started as a Massive Multiplayer Crossover with a handful of original characters, but has slowly evolved to the point that the originals nearly outnumber the crossover characters. To the point that what most fans complain about is not that the Fatal Fury cast have been downplayed in what used to be their tournament, but that one original character (Ash Crimson) has stolen the spotlight from another original (Kyo Kusanagi).
Kyo Kusanagi actually had the spotlight stolen from him years earlier in KOF '99, with K'Dash, who in turn got it stolen by Ash Crimson It seems that every major story arc has a different protagonist, though Ash Crimson is definitely a Villain Protagonist, as well as rather effeminate, which would explain his unpopularity. Until it was revealed that Ash was always on the side of the heroes, but had a very risky plan that relied on him pretending to be a Villain Protagonist and on ultimately causing his own death. A good part of the fandom forgave him.
The 3D Maximum Impact Series take this further by having different protagonists, Alba and Soiree Meira.
Neo Geo Battle Coliseum introduced two protagonists named Yuki and Ai, as well as an antagonist named Goodman. The Orochi clone Mizuchi may also count.
Not to mention Shadow, Metal Zangief, and Sunburned Sakura. Marvel vs. Capcom also had Shadow Lady and Lilith-style Morrigan.
Marvel Vs Capcom 2, where we got Ruby Heart, Amingo, Son-Son's granddaughter (also named Son-Son), and the boss Abyss.
Capcom Vs SNK 2 Mark Of The Millennium with Shin Akuma (as in "God Akuma", not "True Akuma") and God Rugal. These were slight twists on existing characters (Shin Akuma and Omega Rugal), but could only exist as part of the crossover, since their existence involves Rugal absorbing the Dark Hadou or forcing his Orochi power into Akuma.
SNK Vs Capcom SVC Chaos introduced Serious Mr. Karate and Violent Ken. The latter is based on the brainwashed Ken from The Animated Movie.
Namco X Capcom, starring Reiji Arisu and Xiaomu, as well as Reiji's arch-nemesis Saya, all tied together by dimension-hopping shenanigans that gave an excuse for all the game settings (ranging from Ghosts N Goblins to Street Fighter to Soulcalibur to Xenosaga and even more) to be stitched together without too much scarring.
Cross Edge hosts a whole slew of new characters that are not from any of the games of the five companies that "Cross" it, including lead characters Kannagi Yuuto/York Neely and Aiba Mikoto/Miko Aiba. There's also Toya/Troy, Anesha, Judas, Lazarus, Augustine, Raizen, Vivi, Eruma/Mimi, and Iruma/Cece, who all seem to be part of a greater plot of gathering different "Souls" from different worlds, thus explaining why you see characters from Darkstalkers, Ar tonelico, the Atelier Series, Spectral Souls, and Disgaea.
Ingrid from Capcom Fighting Jam, although she was eventually added into the PSP version of Street Fighter Alpha 3. She was supposed to debut in the canceled 3D fighting game Capcom Fighting All-Stars with two other characters: D.D. and Rook.
Shin from Street Fighter Online: Mouse Generation is the only character in the game who isn't from a prior Capcom game or another license.
The Super Smash Bros series had a few; Master Hand and Crazy Hand, the Fighting Polygons/Wire Frames/Alloys, and Giga Bowser. Brawl brought in the all-original Subspace Army with its leader Tabuu. Notably, the playable characters are still entirely crossovers.
The Subspace Emissary also brought us the Ancient Minister, who's actually a subversion. He's really R.O.B.
Tales has the crossover series Tales Of The World: Narikiri Dungeon. The first game, on the Game Boy Color, had Mel and Dio, an entirely too cute matched set of silver-haired chibis with the ability to dress up as classes or monsters. When the series made the jump to the GBA, the new protagonists were given the ability to assume generic classes... or to go the whole way and become the other characters they were fighting alongside. These also included villains, spirits, and cameos like Wada Don and Mr. Driller. Oddly, Pac-Man has not been a costume yet.
The first one was not a crossover, it was a sequel to Tales of Phantasia (and the title was even Tales of Phantasia: Narikiri Dungeon). The crossover began with Narikiri Dungeon 2, and was when the Tales of the World title started being used.
And with the Tales Of The World: Radiant Mythology series, the three Kanonnos and the three flying creatures (Mormo, Paneer, and Rockspring), among other original characters.
Marvel Nemesis: Rise of the Imperfects had half the cast as originals, the titular Imperfects, who were also to become Canon Immigrants. They didn't work out so well.
Mortal Kombat Vs DC Universe has Dark Khan, a Bigger Bad who could only be created due to the crossover, being the result of Darkseid and Shao Kahn being struck down by Superman and Raiden respectively as they were going to escape into dimensional portals while the universes were seperate. All this happening at the exact same spot in time and space, thus merging them and the two universes.
The Nicktoons Unite series has the Wise Old Crab as a rare case of a supporting original character (all of the playable characters are crossovers) and the Mawgu and Globulous Maximus as Big Bads (the other villains in the series are crossovers).
Takumi and Haruna of the Shin'i Organization and villainess Kuu in Battle Moon Wars.
The upcoming .hack game .hack//Link features characters from many previous entries in the series (Kite, Pi, Haseo, Midori, Albreio and so on) due to the fact that this is a 3rd version of the The World. The main character and villains however are entirely new.
Marvel Ultimate Alliance 2 follows the rough plot of Civil War but eventually split offs into an original storyline when the nanites go rogue, creating a new villainous, viralHive Mind intelligence known as the Fold. An evil cyborg Nick Fury being the closest thing the infection has for a figurehead effectively means they're more of a Original Generation concept.
SD Gundam G Generation has a variety of original units which are Mobile Suit Variations (Prototypes which are handwaved to not exist in the anime). But the most prominent one is the Phoenix series of mobile suits which the Phoenix Gundam is said to be the strongest Gundam even in comparison to ∀ Gundam. Along with that is a set of original pilots which either is just meant to be filler or is part of a side story plot.
G Generation Wars play with this trope. You won't even aware of its existence until you unlock an EX mission, and it only appear in last two missions. Yet it turn out that events in the game was controled by Generation System super computer.
Dissidia: Final Fantasy has Cosmos as the Big Good and the guiding force driving the heroes to defeat Chaos. Chaos himself sorta counts as this too, as he's from Final Fantasy I originally, but has been reinventedtothepoint the only real similarities between his original incarnation and his Dissidia incarnation is their physical appearances and name. The Final Fantasy I Chaos was a generic demon One-Winged Angel transformation for Garland—Dissidia's Chaos is a Physical God and the Bigger Bad to all previous Big Bads.
Note that it's still implied that he is in fact the Chaos from Final Fantasy I seeing as rigorous time-and-dimension travel is explained to come into play through the reports in 012.
Garland says that he and Chaos are the same thing in a cutsceen anyway.
The Mario Golf and Mario Tennis series included original characters in the N64 Mario Golf, GBC Mario Golf and Tennis, and the GBA Mario Golf and Tennis versions, so that players would have a decent selection of characters to choose between in the former, and enough rivals and opponents, as well as as Playable OC Stand Ins, in the latter four. Both GBC games can transfer their playable characters and their stats to their respective N64 versions, and the GBA and GC Mario Golf games follow in their footsteps.
The Compati Hero Series, a Banpresto video game series which crossed Super-Deformed versions of Ultraman, Kamen Rider and Gundam, introduced a fourth hero in The Great Battle action game series called Fighter Roar. Roar is not a hero from any previous franchise, but a new character meant to represent Banpresto.
The Parodius games have always had a greater or lesser number of Konami charaters in their player character roster. In the initial games the only character unique to the series was the octopus known as Takosuke in later games, but from Gokujou Parodius onward the number of original characters, not counting Expys and Palette Swaps, grew steadily larger.
Rockman Crossover, a iOS social game developed to commemorate the 25th anniversary of the Rockman franchise, centers around a new Rockman named "Over 1" who must rescue the other seven Rockman heroes.
Princess Lana and Kevin of Captain N: The Game Master serve two purposes. Princess Lana's storyline exists to create a single universe for all the crossovers to take place in along with a central plotline for the series to follow. Kevin exists to make sure that A) none of the other franchise characters is the star (which could piss off a licensee) and B) To make sure someone appealed to the target audience of game buying kids. Normally he would also be there to ensure nobody's character got derailed, but... well...
Similar to Reptil, Evan Daniels/Spyke and Kaldur/Aqualad in their respectiveshows are series original characters, both with a number of similarities. They're both African Americans with blond hair, that were indirectly brought into the comics later but recived voal hatred from sections of the fanbase for reasons other than being series original. And, before all of them, Firestar in Spider-Man and His Amazing Friends, created because they couldn't get the copyright to use Johnny Storm.