Originally, Jim Davis created Lyman so that Jon would have a human friend, a sort of sounding board. As Garfield evolved into a more human-like character and began taking over that role, Lyman became redundant.
Garfield's Scary Scavenger Hunt on the official Web site features Lyman trapped in the dungeon, and in the shower (the sequel replaces him with Jon) of a Haunted House. After Lyman is fed a muffin, he mysteriously vanishes. He also runs the bookstore on the site.
Garfield has, in fact, become steadily less reliant on its supporting cast overall, a far cry from the days of the 1980s and '90s when Jon and Garfield were depicted as having fairly active lives outside of the home. Characters such as Jon's family, Irma at the diner, and Garfield's girlfriend, Arlene, have all steadily phased into near-obscurity. Even Odie's appearances are now noteworthy events. The spiders Garfield harasses (and vice versa) have strangely emerged as the strip's most dominant supporting characters in the 2000s.
The mystery was finally solved on a four-part episode of The Garfield Show, called "Long Lost Lyman"!
However, due to the fact that the show is not really linked to the comics themselves, this might/might not be canon.
The eponymous character of Cathy originally had a doggedly determined boyfriend, Emerson, who pursued Cathy with the same hopeless determination as she gave to Irving. He faded away after the first year or so.
In FoxTrot, Peter's girlfriend, Denise Russo, whom he met in the first year of the comic strip, suddenly disappeared in the middle of the decade (a couple of years before it went Sundays-only), but he didn't seem to break up with her (one could say it happened off screen, but at the same time we never see Peter trying to date any other girls).
And once the strip became Sundays-only in 2007, pretty much all of the supporting cast either disappeared or made smaller appearances, leaving just the titular Fox family and Jason's friend Marcus.
Shermy from Peanuts was the first character to get a speaking line in the strip, and Charlie Brown's best friend early on, but then vanished without a trace sometime in the 1960s.
MAD pointed out Shermy's disappearance with "Will Success Spoil Charlie Brown?", a feature they ran several years later in which he comes back to the strip and finds everyone in it has let stardom go to their heads.
Several other Peanuts characters disappeared for the same reason, as creator Charles Schulz has admitted - they just weren't that interesting. The roster of the eventually-missing also includes Violet and the first Patty (the two random Mean Girls in the early strips) - in other words, all the initial characters except Charlie Brown and Snoopy - as well as one-note types like Frieda. Shermy was explicitly replaced by Franklin, the strip's first black kid.
He also did this to a lot of "experimental" characters that never quite worked out. Schulz once introduced a boy who was named 5, who claimed his family had changed their names to numbers, claiming that numbers were becoming more important than names these days. Unfortunately, he was sort of a one-joke character, so he eventually disappeared and was forgotten about.
After appearing in about two strips, Charlotte vanished due to all the criticism she got for going too far in her cruelty. Unlike Shermy, Patty and Violet, who at least were Demoted to Extra for a while, Charlotte never appeared again at all. Charlotte had a lot of potential, too; her full name was "Charlotte Braun," and, as that might suggest, she was created as a female counterpoint to Charlie Brown. She was basically the opposite of him: abrasive instead of timid, over-confident instead of self-loathing, convinced the world owed her instead of convinced the world hated her. Schulz quickly learned that people liked Charlie Brown for the same reasons they hated Charlotte. He replied to one letter writer who asked him to take her out of the comic, "I will remove her, but how do you feel about causing the death of an innocent child?" (the letter included a picture of Charlotte with an axe in her head). Ouch.
Frieda (apart from being an example herself) also had a cat named Faron, who was featured prominently for a few months and then disappeared forever. Word of God says this was because it wouldn't make any sense for her and Snoopy to interact as they could only communicate through thought bubbles. Charles Schulz apparently got over this as Snoopy's siblings made later appearances, and he communicated with them through thought bubbles.
All of these characters are featured in the iOS app Snoopy's Street Fair, along with some of the more obscure ones, such as Thibault, Lydia and Joe Agate.
Just about every other character in the Prickly City strip besides Carmen and Winslow was a victim of this.
This was the case with the main characters in Out Of The Gene Pool. The strip was originally about Rufus and his friends and families. Later in the run, the cartoonist switched focus to Rufus's brother-in-law and changed the title to Single And Looking. Rufus was never heard of again until a year later, when he appeared in the very last strip, snarking about how he never had a proper closure.
In the comic strip Robotman, the titular character got abducted by aliens, and was never mentioned again. Eventually, the comic strip got renamed to Monty, one of the other characters who became the new main character.
Also, the strip was first about him staying with a traditional family, and much of the comedy was based on "weird urban alien hijinks" similar to E.T. or ALF. They vanished with no explanation. This was later lampshaded.
Calvin of Calvin and Hobbes was introduced to an Uncle Max, his father's younger brother, fairly early in the comic strip's run. Max's single visit to Calvin's home was also his last, and as Watterson put it, "Max is gone." He's said that he thought inventing Max would offer new story opportunities, but realized that introducing an extra adult just intruded upon the strip's childhood setting; it'd be difficult to have Max and Calvin's parents interact without branching them off into more grown-up stories, and having Max never refer to Calvin's parents by name (a cardinal rule for the strip) felt too contrived.
Chip Dunham's Overboard had several pirate characters early on that just sort of disappeared over time. To replace them, he's increasingly relied on Talking Animal characters (dogs, mice, and rabbits) to serve as foils to lead character Captain Crow and remaining shipmates Charley and Nate.
Pretty much every character from the early days of Bloom County. Milo was the only character to last the entire run, and before it was over, even he was Demoted to Extra. Notable vanishings were Milo's grandfather and Cutter John's girlfriend Bobbi Harlow.
Which is weird because Bobbi was already well established as a foil for Steve Dallas when Cutter John was introduced to be her boyfriend.
The original premise of the strip was about Milo living in a large boarding house full of quirky residents. All of these people ended up vanishing, and eventually, so did the house itself.
Berke Breathed admitted that he hadn't found the strip's "center" in early comics; when Opus emerged as that center, he took focus and there was no longer a need to "try people out".
In Beetle Bailey, Beetle had a steady girlfriend named Bunny, and they had been together as the comic progressed all the way up until the turn of the century, when Bunny vanished without any explanation. It took a few years but Beetle got a new girlfriend, the famous Miss Buxley.
Before he joined the Army, Beetle's girlfriend was Buzz. In 2010, a strip revealed that she was now dating his old friend Bill, another character who disappeared when it switched from college humor to military humor.
There was a homeless match seller who appeared in a number of early Alex strips. He vanished a few years into the run without explanation.
Barney Google. The star of his own comic strip from its creation in 1919, he gradually faded out in the 1950s and hasn't been seen in over half a century - but STILL gets top billing in the Artifact Title of the strip, Barney Google and Snuffy Smith. Most people just know the strip as Snuffy Smith nowadays. Oddly, the strip also went through a stretch where it was entitled Barney Google and Spark Plug, the latter being a racehorse who was also dropped in the transition to Snuffy Smith. Though Barney Google did return in 2012 as a short cameo.
In The Amazing Spider-Man, Maria Lopez was a professional rival and potential romantic interest for J. Jonah Jameson. After a reboot storyline that was later retconned into a dream sequence, Maria had vanished without explanation and has never been seen since.
Early in Pearls Before Swine, Pig and Rat got a new roommate, Leonard. The creator of the strip was hoping he'd make for some good material, but it didn't work out and he was rarely seen. Eventually he was killed off screen, getting his head stuck in the toilet and drowning.
In Piranha Club Ernie was going to Doris Husselmeyer's house to meet her for the first time. Another man was there too, and Ernie asked whether he was going to meet Doris as well. The man said that he wouldn't, and that he was going to meet "the pretty sister," who, at the end of the strip, turned out to be horribly ugly. She was never seen again.
There was once a strip called Thimble Theater, about a woman named Olive Oyl, her equally punny named family, and her boyfriend, Harold Hamgravy. Ten years in, it introduced a sailor named Popeye who proved to be an Ensemble Dark Horse, and shortly after that most of Olive's original supporting cast disappeared.
Alison Bechdel has frequently noted just how tough making Dykes To Watch Out For was, so it's not surprising that she just plain gave up on a number of characters:
Harriet - Originally Mo's long-suffering girlfriend, Bechdel liked her enough to keep her around long after the breakup, and she transitioned to a viable secondary character. However, new characters (particularly Stuart, Jiao Raizel, Jonas, and Cynthia) and their storylines eventually took top billing, and Harriet was gradually phased out.
Naomi - Apparently, Bechdel intended her to be part of the main cast and just could never find her niche. She did a few minor things (most notably the Passover strip), was revealed to be bisexual, and finally had one or two throwaway cameos before never being mentioned again.
Thea - Handicapped (but doesn't want to make a big deal out of it), femmey, somewhat rough personality, used to be with Sydney, now with Maxine (apparently the longest-lasting, most stable couple of all). That's it. Bechdel even stated that Thea's an artist but could just never work it into the strip. Vanishes almost immediately after Madwimmin Books closes. Lampshaded in the "strike" strip:
"I thought I would be a nice two-dimensional character like the rest of you. But noooo, I just show up on my crutches every tenth episode like a damn poster child!"
Jezanna - One of the integral characters for much of the strip's run; unfortunately, she lost all relevance after the bookstore closed. Got a handful of strips and then vanished.
June - Much like Thea, she simply got lost in the shuffle. She left the strip immediately after breaking up with Sparrow and only appeared in a few cameos afterward.
Yoshiko - Ginger's artistically minded student (and apparently a longtime friend of Lois as well). Yoshiko was actually listed as a main character in one of the books. She never did anything of note and disappeared after a few strips.
Sophie - Madwimmin Books intern who had a couple of witty verbal jousts with Mo. She was another victim of the bookstore's closing.
Early on, Mother Goose in Mother Goose and Grimm lived with a male pig named Ham. He was quietly dropped over time.
British comic strip The Perishers has an interesting story. One recurring character, Kilroy the tortoise, believed himself to be the reincarnation of Adolf Hitler, wore a swastika around his neck, and spoke with a mangled German accent. In 1984, the newspaper that carried the strip, the Daily Mirror, was purchased by Robert Maxwell - a Czech from a Jewish family, who lost loved ones to Auschwitz. For obvious reasons, Kilroy was never seen again.