This is kind of hard - from experience, it's easier to nominate characters I started liking, but then stopped. But I guess I can nominate:
-A Song of Ice and Fire: Jaime Lannister. Goes from one of the most despicable characters in the setting to be one of the most interesting.
-Andromeda: Trance, in that she starts out in the "perky annoying" end of the spectrum, but grew on me by the start of season 2. Even "Trance 2.0" managed to keep enough good will.
-Avatar: The Last Airbender: Maybe Zuko? I dunno, I kinda liked him from the start, but he does have the most dynamic character arc in the series.
-Babylon 5: John Sheridan, in as much that come season 2, I'm thinking "well, I knew this would happen, but I miss Sinclair." By the end of season 2, I liked his character more.
-Continuum: Maybe Betty? I mean, she was never that major of a character, but at the least, she does stark in the generic "office crush" zone the viewer is aware of while focusing on Kiera and Carlos. Still, her death was brutal, and by said death, it's shown that while she did work with Liber8, she did so by seeing what was happening to the VCPD. I feel her arc was undermined by how Continuum never really executed its concepts all that well IMO, but hey, I'm reaching for examples here.
-Dark Matter: Five/Das...sort of. I mean, I never really disliked her, but at the start of the series, she's clearly the odd one out among the Raza. By the end of season 1, she might be my favorite member of the crew by virtue of that fact. That in a crew full of backstabbers and killers, she's just a thief who was in the wrong place at the wrong time.
-Darksiders: I guess Uriel. She starts in the territory of "holier than thou angel who wants War dead because she had a crush on Abaddon." By the end, she might be the only moral character left in the setting (and no, I don't consider War to be a "good" person, and he's not that interesting a character either)
-Dead Space: I guess Zach. He starts out in the context of just telling Isaac what to do, and how to do it, as you struggle to keep the Ishimura afloat. But when he died, well, that wasn't nice.
-Diablo: Tyrael. He's an arse in Diablo II, and even though he's fleshed out in The Sin War, he remains an arse for half the time, and only stops being an arse by the end of his character arc (and in Demonsbane he's still an arse). By the end of Diablo III, he's now my favorite character in the setting.
-Farscape: Ka D'Argo. When we first see him, he's the sterotypical "I'm angry and a warrior alien, and I hate John because he looks like a sebacean." By Peacekeeper Wars and his death...well, yes, that moved me. But hey, what better way to go out?
-Final Fantasy: Yuna (only played FFX). By the start she's passive to the point of irritation. Tidus goes on a similar arc, but I feel that Yuna's is more pronounced, and does fit the criteria of a character I didn't like at the start.
-Golden Sun: I guess Felix, but that's kind of cheating, given the revelations The Lost Age presents. I guess I never really 'disliked' Felix per se, since most of my ire was directed to Saturos and Menardi.
-Halo: Bornstellar...sort of. At the least, as the IsoDidact he becomes the character I knew from Halo 3, rather than the whiny little twat he was at the start of the Forerunner Saga.
-Harry Potter: Snape. I hesitate to call him a 'good' character (in terms of morality), but by the end of the series, he's certainly a sympathetic, multi-layered one.
-The Legend of Zelda: Midna. Perhaps the definitive example I can think of. When we first see her, she's rude, irritating, snide, etc. When she gets blasted by the Light spirit and I'm carrying her around to find help, I'm thinking two things - 1, "please don't die, please don't die!. 2, "wait, when did I start caring about you this much?" Midna is a character that I not only grew to love, but did so without me even noticing until she was at death's door.
-Merlin: Arthur. In that his entire arc is based on going from spoiled, arrogant prince, to "the once and future king." Botched ending aside, I'd say that worked.
-Mistborn: Rashek. Goes from one of my most despised characters to one of the most sympathetic by the trilogy's end.
-Sonic the Hedgehog: Amy Rose...sort of. In that every incarnation prior to her "Boomverse" version was vapid in one way or another, defined only by her love for Sonic, or in STC's case, by "girl power!" But Sonic Boom (the cartoon) actually made her feel like a character. Not that deep of a character, but still a character with interests, perks, cons, and whatnot.
I guess if that's cheating, I can nominate Marine.
-Terminator: I guess maybe the jump from T1 to T2, if you count both T-800 versions as the same character?
-The Walking Dead: Hmm...there's not really any shortage from hate-love or love-hate in this case. I guess I can nominate Beth. Season 2, her only action of note is her attempted suicide attempt, and it gets limited sympathy from me given what Rick and co. have had to go through. Season 3 she's a bit more pro-active, but comes into her own in season 4, where she feels like she's taken a place in the foreground - hasn't lost her season 2 character, but has certainly grown. And by season 5...well, I know what's coming (thanks Internet), but at the least, I'm rooting for her.
-Whoniverse: The Tenth Doctor, but this is kind of cheating. I was first introduced to the series with "nu-Who," so when Nine regenerated into Ten, I had no idea that there was a precedent for this, and stopped watching. Catching Ten on TV once, I gave him a try, and he now stands as my favorite Doctor incarnation. So, go figure.
-Wing Commander: Tolwynn. As in, he starts out as nothing more than an arse in WC2, and remains an arse in WC3, but a likable arse thanks to Malcolm McDowell. Come WC4, he's a genocidal arse, but still, an interesting arse. So, I guess that might count.
-Xenoblade: Melia. When we first see her, she's a sterotypical uppity princess character who basically forces the plot to stop for the Alcamoth 'stuff,' and, well, yeah. Still, by the end of it, I quite liked her. She gets perhaps the most raw deal out of any of the main characters, but she still hangs in there until the end. Also, I suppose Egil could come as runner up, but he's too ambiguous for most of his arc for me to actively dislike, even if he does go out like a hero.
-Xenopedia: Grego. Make no mistake, Grego is an arsehole. But he's at least an arsehole that goes from abandoning his own crew, to killing his own crew, to helping a yautja by virtue of being an arsehole. He's the kind of arsehole I can't help but like once he becomes the main character in the Predator arc of 'Fire and Stone'.