This is a source of conflict for Kaoru in Amagami when her single mother starts dating again.
In the Anime/Manga Midori Days, Seiji helps his neighbor Shiori to accept her new mother after Shiori's initial dislike.
Conversed in Lucky Star when Sojiro has a discussion with his daughter Konata about him possibly getting remarried. Sojiro anticipates that Konata would be against it because her mother Kanata would always be her mother, but to his confusion and embarrasment Konata apparently defies the drama by saying she'd be totally cool with it and even asks when she'd get to meet her new mom.
Odd variation in Marmalade Boy. Miki's parents Jin and Rumi get divorced amiably, so do Yuu's parents Youji and Chiyako... and then they swap couples and the two "new" families move into a Big Fancy House to avoid parental issues. It takes Miki more than a while to accept it, though: despite liking her step-parents very much, she can't get used to the rest of the deal, unlike her nonchalant new stepbrother Yuu. It turns out that Yuu himself was just as distressed as Miki, and for the same reasons. He just hid it much better, and in fact he confesses that he envied Miki for being able to voice her disagreement while he couldn't bring himself to do so.
While Chief Togakushi of Region seems oblivious to the feelings of his coworker Asagiri, his daughter Kanon is quick to notice the way she acts around him. This drives a wedge between the father and daughter, as she feels that it's too soon for him to abandon the memory of her mother.
A good part of Nanako Misonoo's early angst is related to how her mother married her stepdad when she was five. Not because said stepdad is a bad person, far from it (plus she' a total step-daddy's girl), but because her new classmates use it as an excuse for bullying her.
Mariko Shinobu's father mixed it with Your Cheating Heart, as he first cheated on Mariko's mom with an actress and then ran away with her, pretty much abandoning Mariko and her mom. Her classmates, and specially Aya Misaki, also use this to bully her; in fact, in the anime one of Aya's earliest Kick the Dog moments come from first trying to shame Nanako over her family, and when Mariko jumps in defense of her new friend, she reveals Mariko's family situation to the whole school... No wonder Mariko snaps badly on her. It's even WORSE when the Shinobus DO get divorced (very publically), Aya again gloats about it... and Mariko cracks so violently that she attacks her with a boxcutter.
This takes over the plot by the end of Sakende Yaruze! and all reactions are covered. Initially Nakaya and Tenryuu get along (1), but Nakaya is also nervous about older men in general (2), and when he finds out that Tenryuu wants a relationship with his father he is very averse to the idea and starts to dislike him (3). Nakaya moves out once they get together, and it's suggested that while he gets used to Tenryuu in the end he's never entirely happy with his father's choice in a partner.
In Tenchi Muyo! Tenchi's mother Kiyone died when he was young, and even though he's an adult by the time his father Nobuyuki gets around to marrying Rhea, who's known them since she was in high school and was Nobuyuki's assistant in his architecture office, she's effectively already been his mother for years. Their relationship is good enough that Tenchi eventually realizes that many of his memories of his mother are actually composites of Kiyone and Rhea, since Kiyone's personality was actually quitedifferent from what he remembered. As such, Tenchi is more than happy to see them finally get married, especially because it's what Kiyone would have wanted since she and Rhea were friends.
In the Fushigi Yuugi manga, Miaka sees her divorced mother out on a date one afternoon. Later, Miaka finds out that Mrs. Yuuki read her diary, and misinterpreted writings about Tamahome as evidence that Miaka was dating and skipping Cram School classes. Miaka denies this, and also calls her mom out for dating someone yet saying she can't even think about boys. This gets Miaka slapped in the face, and she ends up running away, thus setting the story into motion.
This is implied to be one of the reasons why Asuka didn't live with her father in Neon Genesis Evangelion. She's seen talking on the phone to her stepmother in German (in a conversation full of Bilingual Bonuses in both the English and the first Latin-American Spanish dub) and acts friendly enough, but as soon as the call's over she drops the act and explains to Shinji that, while she doesn't necessarily loathe the lady, she doesn't like her either and doesn't want to even think of her as a replacement for her dead mother, Kyouko. It's made worse worse by the implication that Asuka's stepmother was both the doctor that treated Asuka's maddenedMissing Mom while alsobeing the 'other woman' in the Sohryus' decaying marriage. For even worse, there's a further implication that Kyoko's suicide (which in reality was an attempted Murder-Suicide where she hung the Creepy Doll she believed was Asuka in her madness along with herself) was at least in part to spite her husband.
In Kyo Kara Maoh!, flashbacks show that Gwendal had a type 3 relationship with Conrad's human father Dan Hiri, partly due to personality conflicts (Conrad's dad was something of a Boisterous Bruiser, while Gwendal was The Stoic and a bit of a grouch even as a kid) and partly due to Fantastic Racism between Mazoku and humans. In the end, it's shown, Gwendal's hatred for him and desire to surpass him left a permanent mark on Gwendal's personality...and Dan Hiri considers having made such an impression on such a great man to be the legacy he spent his life searching for.
In Howl's Moving Castle, Sophie and her mother have a good relationship but Sophie never gets to see her mother's new husband.
Used tragically in Death Parade. Yousuke's mother left him at a young age and his father remarried later. His step-mother said it was one of her dreams for him to call her "Mom" but Yousuke didn't take well to it. His last memory was his step-mother crying over his body after he jumped from his bedroom window. After remembering that he committed suicide, and for such petty reasons, Yousuke breaks down into tears.
In Aishiteruze Baby there's Kokoro's father, who eventually remarried after the death of his wife. Kokoro doesn't live with either him or his new wife but in a a rather nice apartment to give them some space. And later in the manga, the recently widowed Miyako is revealed to be dating again. Her niece Reiko does not take it well.
Magical Girl Raising Project has this come up multiple times as part of Kano Sazanami's/Ripple's backstory. An extended flashback shows two instances involving it, starting with an incident in elementary school where other kids make fun of her over her last name changing every other month or so (which ends with her punching them out after they suggest that her mother is a whore, and as such she must be one as well), and ending just before the story proper as she meets her mother's newest husband (which ends with her knocking him unconscious and running away to live on her own after he starts trying to molest her).