Meta Knight has been in absolute command of the fandom's attention from his debut, not only as both Dedede's Dragon with an Agenda and The Captain of the Meta-Knights but also as the only other known member of Kirby's species for over a dozen years until Super Star Ultra introduced Galacta Knight. Following his debut, he immediately became an Ascended Extra and the Big Bad of one of Super Star's mini-games, and then a series of major plot roles and occasionally playable appearances (starting with Nightmare in Dreamland), both in his home series and in Super Smash Bros. He's absent from Triple Deluxe, for some reason ( though Dark Meta Knight returns as the final boss of Dededetour), but finally returns in full glory for Planet Robobot.
Rick, Kine, and Coo, the animal friends from Kirby's Dream Land 2, are beloved by the fanbase, enough that in spite of only being part of the gameplay in their debut game and its sequel, they make cameos throughout the series. They even got to appear in the anime in spite of the fact that they were not created by Sakurai.
Nago, Chuchu, Pitch, and Gooey, the animal friends from Kirby's Dream Land 3, are also quite popular for similar reasons as Rick, Kine, and Coo. In particular, Pitch grew on fans for having some of the best copy ability variants in the game, and Chuchu gained some extra leverage for the amount of Ship Tease she had with Kirby. Gooey also has his own group of fans for his quirky design, characterization, and backstory (he's a specimen of Dark Matter who took a Heel–Face Turn and befriended Kirby).
The one-shot characters from Kirby 64, such as Adeleine, Ribbon, and 02 have a TON of fanart for them. 02 may mainly have been popular due to being surprisingly symbolic for a Kirby game (he has an angelic appearance and spits out blood) and due to his awesome battle music. 02 even got a remix of his theme music in Brawl. Ribbon's popularity is probably due to her implied relationship with Kirby. Adeleine became a recurring character in the webcomic Brawl in the Family.
Bandana Dee, who went from one-shot opponent in a Super Star minigame, to the confirmed identity of the Waddle Dee boss in the remake (with speaking parts!), to a playable character in Kirby's Return to Dream Land, is gaining in popularity lately. Some Fanon even holds that he's the same Waddle Dee who partnered with Kirby in Kirby 64, just without his Iconic Item.
In the time between Return to Dream Land and Palnet Robobot he has made more appearances than both Meta Knight and King Dedede.
He's even become a very popular request for Super Smash Bros., and given how important he is nowadays, it's easy to see why some people thought he might have been a candidate to be put in.
Many of the enemies may also qualify, such as Waddle Doo and Knuckle Joe (who might both owe this to the anime).
Sword Knight definitely qualifies. Despite not appearing as often as Blade Knight, he has a large fanbase.
Kirby 64 is a Darkhorse game for the series. Aside from starring the aforementioned characters, it's a great aversion of Polygon Ceiling, opting for a 2.5D presentation that keeps the gameplay unchanged, while providing some of the best animations and background art for the console. It possesses a fair amount of challenge while not being unfairly hard, and the popular mechanic of combining powers hasn't been seen since. Nowadays any game featuring character cameos or remixed music from the game is typically met with enthusiasm by fans.
A lot of fans have noticed that certain characters from the anime have made cameos in Kirby Mass Attack, and fans of said anime naturally squeal with joy upon seeing them again. These characters include Escargoon, Customer Service, Chef Shiitake, and Max Flexer.
The villains Marx and Magolor both seem to qualify. Marx for appearing as the final boss in what is generally regarded as the best game in the series by the fans, playing Kirby like a fiddle up until the climax, and being one of the series' earlier examples of a surprise creepy. Magolor, much like Marx, plays Kirby like his pawn for the entire game, has several awesome theme songs, and ends up actually learning and using Kirby's own Ultra copy powers against him. Magolor ended up becoming so popular that Hal brought him back from the dead, complete with a Heel–Face Turn.
DynaBlade, from the Kirby Super Star minigame of the same name, has a pretty robust fanbase, no doubt due to being a giant rainbow-colored armor-clad bird who moves at sonic speeds and puts up a mean fight as well as simply trying to feed her new family. She helps Kirby board the Halberd in the Revenge Of Meta Knight minigame, and she's had her own mission in Kirby's Air Ride, a trophy and sticker and Ridley borrowing her signature attack in Super Smash Bros Brawl, and two major appearances in the Kirby anime.
Susie from Planet Robobot earned some love for her backstory, Affably Evil nature and being the first villain in the series to help Kirby but not turn good as a result.