The 1996 Star Wars spin-off, Shadows of the Empire (novel, comic book, video game, and breakfast cereal), being set between The Empire Strikes Back and Return of the Jedi, found a replacement character for Han Solo in the lovable scamp Dash Rendar. Dash not only sported Han's in-your-face attitude, he flew a nearly identical ship to the Millennium Falcon and had a wacky robot sidekick. Though hastily offed when he no longer served a purpose, Dash apparently still exists in the hazier reaches of the "Expanded Universe."
Gavin Darklighter, who is the cousin of Biggs Darklighter.
With the repeated "new Dark Lords" and "new superweapons," the WHOLE Expanded Universe should be called Shadows of the Empire!
After Ned Stark's death at the end of A Game of Thrones, Davos Seaworth shows up to serve the function of The Stoic, family and honor oriented, speaker-of-truth-to-power to a Baratheon king. They even mirror each other in interesting ways, particularly their fates ( Ned is killed in a last minute whim of Joffrey's, Davos miraculously survives his ship's sinking) compared to their children's ( Ned's children find ways to thrive in adversity, while four of Davos' sons are killed in battle).
Oberyn "The Red Viper" Martel was Too Cool to Live, and fans have suspected that The Scrappy Gerold "Darkstar" Dayne was created as one of these. Both men are the black sheep of their families and have badass reputations and are known by cool nicknames. More to the point, both men are from the some place and Dayne interacts with Oberyn's family members.
In-universe example: Gregor "The Mountain" Clegane, a huge, insanely strong and brutal thug is supposedly killed at the end of Book 3. In Book 5, a similar-sized knight who refuses to take off his armor called Robert Strong appears. Many characters suspect that Strong is in fact Clegane's zombified corpse.
A meta-example in Neal Stephenson's The Baroque Cycle: Daniel Waterhouse notes that some people can be replaced in their positions with only the most superficial elements changing, but other people are more crucial to the story, and their loss will forever change the status quo no matter who replaces them.
In Death series: Eternity In Death has Eve and Peabody questioning the housekeeper who worked for the murder victim. Then they go to see the murder victim's friend, whose housekeeper might have been a clone of the previous one. The story states that the two housekeepers are sisters.
Laura McCall for Taffy Sinclair in The Fabulous Five series. Even though Taffy is still present, she and the group have called a truce in the final book of the previous series and she has become somewhat nicer. Laura is the new Alpha Bitch and apparently doesn't have any of the Freudian Excuse that Taffy did (overbearing stage mother, etc.)
In The Infernal Devices: Cyril is essentially an older Thomas, brought in to replace him after his death.
In James Ellroy's novel The Big Nowhere, one of the protagonists is Buzz Meeks, a disgraced ex-Dirty Cop who has since gone on to become a crooked Private Detective, Hollywood fixer, and enforcer for Eccentric Millionaire Howard Hughes. Since Meeks was killed off early in L.A. Confidential, his role in Ellroy's Underworld USA trilogy was filled by Pete Bondurant, a character with a virtually identical backstory.
In Elmore Leonard's novel Split Images, Detroit homicide detective Bryan Hurd is an obvious stand-in for Detroit homicide detective Raymond Cruz. Cruz appeared in some of Leonard's prior work, which was being optioned at the time.
In-universe example in Vampire Academy. When Dimitri turned into a Stringoi and Rose quit her training, Lissa needed new guardians. Grant was a seasoned guardian and trainer, similar to Dimitri. Serena was a girl close to Lissa's age, similar to Rose.