As names are for people in real life, they are an integral part of the identities of fictional characters. They will usually be made up of at least a given name and a last name linking them to one part of their family. Therefore, you can count on any person who wants to disassociate themself from any family member sharing it by doffing the name that links them. In most cases, this family member is the father, due to long-standing traditions involving the woman's adoption of her husband's surname and the bestowal of the father's surname to all of his children. Unlike your standard Meaningful Rename, which is usually part of the story's narrative and its significance derived from the meaning of the new name, this practice is marked by what is renounced: the part of the name linking the person to his or her despised family member.
This can be done in a variety of forms. In cases where this issue comes up in The Maiden Name Debate, a much bigger deal will be made over a man deciding to change his name as part of the marriage than it would if a woman were to do the same due to the fact that a woman is usually expected to change her name in any case. And speaking of maiden names, a common practice among characters who do this is to take their mother's maiden name (or other equivalent) after renouncing the surname they got from their father. Others still will take an entirely different name or give themselves an Awesome McCoolname. In rare cases, the person renaming themselves will decide to go around with Only One Name.
Subtrope of Meaningful Rename. Often overlaps with Nom de Mom.
Anime & Manga
Oishinbo: The male lead holds his Insufferable Genius father responsible for the death of his mother, so uses her family name instead of his father's.
Tegami Bachi: Largo takes his mother's maiden name of Lloyd after leaving Kagerou.
In the Queen's Blade franchise, the main heroine Leina Vance is never refered as such by herself, since she doesn't want to be associated with nobility and her family anymore, not to mention in the sequel, she becomes a masked fugitive, so she doesn't want to be related with her sister Claudette, the new queen.
One Piece: Ace uses his mother's family name Portgas rather than his father's Gol. His reasons for doing so are complicated and sad.
In Y: The Last Man, Dr. Mann took her name from Mann's Chinese Theatre to spite her father, who was a jerk and accidentally caused the gendercide in the process of sabotaging her work.
In the Silver Age, Lex Luthor's father disowned him and changed the family name to Thorul. This came to light when Lex's long-lost baby sister turned up.
In Diabolik, inspector Ginko brought this Up to Eleven: everyone calls him Ginko because he shed both his family and his personal name to distance himself from his father upon discovering he was a corrupt judge. Nobody knows neither his original name nor the origins of the new one.
In Warrior, Tommy fights under his mother's name, Riordan, rather than his father's name of Conlan. Partly, this is because he's a deserter from the army and doesn't want to be discovered, but his relationship with his father is sufficiently bad that this trope can still apply.
The Count of Monte Cristo: Villefort, a Royalist, changed his name to disassociate himself from his Bonapartist father, Noirtier.
Albert de Morcerf does this after he finds out what a bastard his father was and his mother even suggests to him that he take her maiden name instead.
In the Spaceforce series, Jez doesn’t use her ‘House’ name, Aznata (although she did give it when she was presented to the Taysan Empress in the second book). She implies that it’s because she fears attack from the Scree, who massacred most of her people and targeted nobles in particular.
Throughout the Harry Potter series, Lord Voldemort, né Tom Marvolo Riddle, repeatedly disparages the given and last names that he got from his filthy Mudblood father. Both Dumbledore and Harry refer to him by his real name just to make a point.
Jon Cade, the protagonist from the series The Jerusalem Man by David Gemmell, changed his surname to Shannow to distance himself from his brother Daniel Cade, a notorious criminal.
Derek Harris from the AuntDimity series so despised his father that he changed his name; he was born Anthony Evelyn Armstrong Seton, Viscount Hailesham. The change is also meaningful since he took the name of the estate's carpenter and he prefers to work with his hands, which is is considered unsuitable for a peer's son.
In UnWholly, Lev Calder decides to change his surname to Garrity to honor the memory of his recently-killed Parental Substitute after being more or less abandoned by his entire family.
Becoming Anna by Anna J. Michener is a memoir detailing the years of abuse at the hands of her family members that led to her decision to shed her original forename and surname.
The events of The Ends of the Circle (second book in the Pelbar Cycle) are kicked off by the deliberate mistreatment of Stel by his wife's family. In the next book, Stel and Ahroe have renounced Ahroe's family name.
In Legends of Dune, after Xavier Harkonnen performs a Heroic Sacrifice to kill the evil Grand Patriarch Iblis Ginjo, Ginjo's wife labels him a traitor and elevates her late husband to martyrdom. The stigma on the Harkonnen name results in Xavier's children adopting their mother's name Butler. One of them, Wandra, marries Quentin Vigar, who then changes his name to Butler. Inverted by Quentin's son Abulurd, who finds out the truth about his grandfather and changes his name to Harkonnen despite his father's protests.
In Flowers in the Attic, Chris Sr. and Corrine changed the family surname from Foxworth to Dollanganger after being disowned for their incestuous marriage. When Corrine's children escape from her after she poisons them, killing Cory Chris disassociates himself from her by taking the name of Sheffield from the man who subsequently adopted them.
Live Action TV
Paul Stenbeck from As the World Turns changed his last name to Ryan (his Mom's maiden name) to distance himself from his villianous father James.
Dana Brody on Homeland changes her last name to her mother's maiden name after her father is believed to be responsible for the Langley bombing.
The quarian race in the Mass Effect series have both a clan name and a spaceship name, indicating which spaceship community a quarian belongs to. In Mass Effect 2, Tali'Zorah vas Neema (meaning "of ship Neema") is stripped of her quarian "surname" and renamed to "Tali'Zorah vas Normandy" (the name of the player's ship) when she is suspected of high treason and thus subject to exile. Even if she is cleared of charges, the renaming is not reversed, since her bond with Shepard and the Normandy is much greater than with any quarian ship at the time, anyway. In Mass Effect 3, she comments on the irony of "vas Normandy" being originally intended as a badge of shame but the Normandy later becoming the ship that gives the quarian race their homeworld back.
Something Positive: Jason takes Audrey's name when they get married because he hates his father.
Homestuck: In the rebooted kids' universe, Jade Crocker left her abusive adopted (grand)mother and took the name English.
Dresden Codak: Kimiko Kusanagi took the name Kim Ross to disassociate herself from her father who abandoned them in pursuit of his research.
The Simpsons: Lisa takes her mother's maiden name after Homer bets against her in a crossword puzzle competition.
Bobbi Kristina Brown announced her intention to change her name to Kristina Houston specifically to distance herself from her father, Bobby Brown.
Jon Stewart has cited, along with an emcee's mispronunciation of his original surname, "some leftover resentment at [his] family" for his changing his surname from Leibowitz to Stewart. Indeed, Jon Stewart and his father are estranged from each other to this day.
After Oscar Wilde was convicted for "gross indecency", his wife did this on their sons' behalf. She not only changed her and her sons' surname to Holland, a name from her own heritage, to disassociate them from Wilde, but she and her family did everything possible to ensure that Wilde never saw his sons again, even after he got out of prison.
William Patrick Stuart-Houston was born William Patrick Hitler. He was the son of Alois Hitler, Jr. and a nephew of Adolf Hitler himself. Yeah, you can guess why he changed his name.
Nathaniel Hawthorne wrote a lot about Puritans, and almost never portrayed that society as a whole sympathetically. He was the descendant of a judge involved in the Salem Witch Trials, and he hated that. So he changed his last name from "Hathorne" to "Hawthorne."
Inverted amongst sexual minorities or defectors of tightly-knit religious groups. Sometimes parents force the children to assume a different family name to make sure they no more have to deal with children who have brought shame upon them by being deviant and who have defamed the family name. A subtrope of Honor Before Reason trope, and may act as substitute for honor killing.
Also inverted in the case of the actor Jeremy Brett (best known for playing Sherlock Holmes in the Granada TV series). His father, who was of a very high social status (including being a Lord Lieutenant—a personal representative of the British monarch—and an heir to the Cadbury fortune). Brett's father insisted that Jeremy Brett change his name (from Peter Jeremy William Huggins) for the sake of the family honor. (He took the name Brett from "Brett & Co.," the label of his first suit.)
An entire country was forced into this in 1930s and 1940s, as the Mongolian communist party banned the use of surnames (or, technically, clan names) as a relic of the feudal past. Since 2000s, Mongolians started using surnames again, but as the original names were lost, many used this as the opportunity at a Meaningful Rename. Many took up the name of Genghis Khan's clan, the Borjigiin.