The artist Vincent van Gogh was named after a brother who'd died one year before his own birth.
A documentary about the children of Holocaust survivors (Jews, BBC Four, June 2008) featured a woman who had survived Auschwitz but whose young daughter had been gassed there. Later, she settled in Britain, remarried, and had another daughter, who was named after the first one.
Peter Sellers' birth name was Richard Henry, but his parents nicknamed him Peter, the name of their short-lived first child. Eventually, he adopted that name as his own. This has not passed without ironic comment, given his later claims that he could only be his characters and never himself.
This practice was actually used as a plot point in Beethoven's Last Night.
Alexander Hamilton's eldest and youngest sons were named Philip — the younger Philip was born less than a year after his brother died in a duel.
Current Ashkenazi Jewish custom (the superstitious ones anyway) is to avoid naming children after people who died young, or at least not using it as their primary name, out of fear it would bring bad luck and cause the new child to also die young. However, the main tradition is to never name a child after someone who's alive (hence why you don't get too many Jewish Juniors), so naming a child after a deceased child is not out of the question. However, in most cases, the naming-for-the-deceased custom is so common that this trope is often averted.
Naming children after family who died in the Holocaust is very common; many children and grandchildren (and great-grandchildren) of Holocaust survivors end up with extremely long names as living memorials. There's a reason why it's been clinically shown that the descendants of Shoah survivors often have what's been called "PTSD by proxy".
Totally subverted in Mongolian culture where, if a couple's children kept on dying young, they would name the newborn something like 'Vicious Dog', 'Not This One', 'No Name', 'Not a Human Being', or give a female name to a boy. This was to make the evil spirits leave them alone or to confuse the spirits.
Back when infant mortality was high, if a man wanted a "junior," he had to name the first born boy "Myself, jr.", and if that boy died, then the next boy born after the first junior's death would get named "Myself, jr." as well.
In Maus, Art Spiegelman touches on this point, describing some warped sibling rivalry he had with his brother Richieu, who died in the Holocaust before he was born. Specifically because his parents kept a large photograph of Richieu and then, of course, Art's own father calls him Richieu at the end of the book.
Ben Moody, co-founder of the band Evanescence, left the band in the middle of a tour after a falling out with co-founder (and childhood friend) Amy Lee◊. Before long, Moody created a sound-alike band named We Are The Fallen (based on one of Evansecence's album names) with American Idol contestant Carly Smithson◊.
Long-running children's magazine show Blue Peter added a pet dog to the team, which died before the producers had worked out their contingency plan, but after having appeared on screen. They had no plan, and the kiddies would have been confused if it disappeared so soon after arriving, so they did the logical thing and sought out a replacement. When that eventually died, the producers felt that the pet had become familiar and they could deal with it properly.
Some non-pet owners, in an attempt to be nice, commit a massive faux pas when trying to cheer up a friend who has just lost a beloved pet. How? By getting a new one for them that they believe is identical. Trouble is, most pet owners want that one specific pet back, not one that looks like it. Better let them grieve and let them get a new pet on their own.
When Ricky Gervais' talk-show host friend got him a Siamese kitten after his cat died on live tv, he took it surprisingly well and was comforted by how much his friend cared. This is probably because he loves animals and kittens are cute. He named the cat Oliver.
On the other hand, some pet owners deliberately choose a new pet that is different as the similarity would bring back painful memories of the loss of the original.
When George Harrison was still married to Pattie Boyd, Eric Clapton had a brief relationship with her youngest sister Paula, which ended after Paula heard the song "Layla" and realized it was about Pattie.
Director Peter Bogdanovich had a love affair with Playboy Playmate Dorothy Stratten, which ended when Stratten was murdered by her jealous psycho estranged husband. Later, Bogdanovich married Stratten's sister.
The National Hockey League's Winnipeg Jets. In 2011, True North Sports and Entertainment intended to buy the Phoenix (now Arizona) Coyotes and move them to Winnipeg, as the Coyotes were the original Winnipeg Jets, who left Winnipeg in 1996 due to an aging arena and declining Canadian dollar. However, the league was staunchly adamant in keeping the Coyotes in Arizona, so True North went to the next team available for sale, the Atlanta Thrashers. Regardless of which team True North bought, Winnipeggers were happy to have their Jets back.
This is actually somewhat common in North American sports, as the Colorado Rockies, Baltimore Colts, Charlotte Hornets, Cleveland Browns, Vancouver Whitecaps, Portland Times, Seattle Sounders, San Jose Earthquakes, and Florida Marlins were all names after dead teams (with some, like the Rockies, names after a dead franchise in another sport).
Subverted by one soldier whose drone was too damaged for repairs. When told a replacement would arrive, he said he didn't want another one, he wanted Scooby-Doo back.
Dave Barry once wrote about how his daughter Sophie took in a bug from the porch which she named "Melvin". Melvin of course quickly died and he and his wife, instead of explaining this to her, kept replacing it with a new "Melvin" from the porch (there were a lot of those kinds of bugs, apparently). The same column detailed his attempts to buy a pet fish for Sophie which he said had to "look like other fish in case - God forbid - we have to Melvinize it".
The trope namer is actually something that shouldn't be an example anywhere near as often as it is. Well-cared-for goldfish should actually live for decades and reach around a foot in length. The main problem is that people keep them in tanks that are far too small and overcrowded and they suffocate.
The One World Trade Center is this to the original World Trade Center in New York City, which was destroyed in the 9/11 terrorist attacks. This might be one reason the architecture of the center has been so criticized, although many forget the original was also disliked before the attacks (a non-living version of Never Speak Ill of the Dead, perhaps).
Robert B. Parker loved his dog Pearl, a female German Pointer Setter. When she passed away, he promptly bought another German Pointer Setter and named her Pearl and kept repeating this pattern while also behaving as if each Pearl was the same dog (as in the exact same dog and not a reincarnation). His wife Joan joked that if she'd predeceased him, he'd have found an identical woman, gotten her to change her name to "Joan", and gone on as if the original Joan never left.
The Roman Emperor Nero is said to married and castrated the boy Sporus because he (Sporus) looked so much like Nero's beloved late wife Poppaea Sabina.
The Space Shuttle Endeavour was built to replace the Space Shuttle Challenger after the latter was lost in 1986.
◊ Brightman
