Dungeon Keeper Ami features the Unraveller of Mysteries, one of a pantheon of dark gods hostile to Ami (So far she has encountered the god of martial force, the god of undeath/necromancy, the god of vermin; the Unraveller seems to be the dark god of mad scientists). It should come as no surprise that the Unraveller has such a huge following: the author has admitted that she was written with the voice of Gla Dos in mind.
Someone even wrote an epic ballad for him. Not bad for a random squid who shows up out of nowhere and immediately gets killed off.
Friendship is Witchcraft gives us Sweetie Bot, who has become one of the most recognizable characters and even has her own Tumblr.
Yawning Squirtle and Boulder-Dropping Whale in Sonichu, and Mary Lee Walsh, who is frequently the protagonist in various fanworks, and is drawn deliberately with a different design. This is mainly because many find that her real-life basis is actually pretty awesome. Plus, it ticks CWC off.
Bill the Scientist also qualifies for this, since he's also one of the Moon Pals.
Brother-Captain Allen of Tiberium Wars - stoic, calm, quiet, efficient, scary, and perfectly willing to mutilate rapists and then hurl them out windows. Its no surprise that such a bit character has become so liked.
Harry Potter and the Guardian's Light gives us THE INDOMITABLE ARGUS FILCH. In the novels, he's a slightly scary background character. In the movies, he ranges from creepy to disgusting or comic relief. But in this fanfic, his first scene involves him arming himself alongside Hellboy and John Constantine to go into the black forest and fight an army of trolls. Turns out, while he's not a janitor he turns a profit selling enchanted weapons. After he helps demolish the trolls with ease, we next see him tracking Harry down in his invisibility cloak despite not knowing it's Harry, or that he's in an invisibility cloak. Recently, Voldemort invaded the castle with an army of trolls and Filch was first on scene. His response was to ask them for a hall pass. He then ducked around a corner and they traded some fire. He was ordered BY VOLDEMORT to surrender. His response to that was to blow a troll's brains out, and say; "Alright, I surrender, now come a little closer!"It's easy to see why Dumbledore trusts this man to clean his school's floors.
Dragon Ball Z Abridged has Nappa. In the actual series he was a bland Giant Mook that was killed relatively expediently, but in Abridged he is quite possibly the funniest character in the series, and the fans loved him for it, to the point where he got to reappear several times as a ghost.
There is also popularity upon Super Kami Guru, Nail, Dodoria, Zarbon and The Ginyu Force (Specifically Recoome - these guys even got their own opening between Episodes 19 and 23.).
Warriors The Abridged Series (the abridged series of Warrior Cats, not something else), the most popular characters are Bluestar, Redtail, Matthias and Rainstorm. Redtail and Matthias each only appear in one scene. Well, there is Ghost Redtail, but then again, Redtail gained his popularity before that segment. And Ghost Redtail leaves when he realizes Tigerclaw doesn't like him.
While she was included mainly to intentionally avert the Ron the Death Eater treatment the character often gets, Ember in The Legend of Spyro: A New Dawn has been very well received by readers for being a kindhearted, friendly version of the character. This is part of the reason she ended up having A Day in the Limelight chapter were she defeated The Dragon of Empresss Tyrania, the Big Bad of the Gargoyle Story Arc, to save her father herself, instead of ending up a team boss fight as originally intended. This also enabled the writer to give her more character growth and an excuse to Take a Level in Badass, so it worked out in more ways than one.
Bentley the crazy Adorkable hyena from the Rango fanfic Snakes With Guns is easily one of the most lovable OCs in the Rango fandom.
In the The Wizard in the Shadows, Emrys Ap Derfel. Originally created, as Word of God admits, as an unintentional Expy of the author and of Harry himself - green eyes, difficult past, great power and destiny, something which has previously been lampshaded, Emrys Ap Derfel was well received. Several plot developments, an epilogue and the start of a sequel, he's developed into a key member of the main cast and Word of God has hinted that he'll take over Harry's role as The Hero later on in the Expanded Universe.
Eirian and Miriel are given expanded roles in the sequel (the latter had merely been a throwaway reference before hand) and a third, Maglor is pure Spotlight-Stealing Squad.
Yuuka Kazami of Imperfect Metamorphosis was growing to be such a focal point of the plot that the author ended up having to devote several chapters to a giant brawl between her and Yukari to take them out of the direct action for a while and realign the focus to Rin and Rumia. Being a maniacal, sadistic, overpowered gadfly tends to have that effect. Being a Great Old One doesn't hurt much either.
The South of Nowhere fanfic 'Romance on the Set' has one In-Universe on the show Spencer and Ashley star in. Rachel, a character played by Taylor Momsen was intended to be recurring, but gets promoted to regular due to her popularity and even becomes part of the Fan-Preferred Couple with the main character
The Bridge to Terabithia fanfic "Unearthing Secrets" has Conrad, the very British exchange student and Jesse's best guy-friend, due to his realizing Jess and Leslie want to be more than friends before even they do and making frequent and amusing attempts to cockblock Leslie's egotistical Romantic False Lead whenever he is able.
Prince Heins of Frozen Hearts, largely because he cares a great deal for his brother, Hans, unlike the majority of their brothers and even after everything Hans has done.
Donut in Dusk's Dawn, or at least he's the least despised character. Having one of the few somewhat competent voice actors probably helps.