Origins has a list of quirky party members, but it's Duncan who's the character that all the fans of the series love. A big reason for this is because he was Too Cool to Live.
Finn and Ariane, the two companions from the Witch Hunt DLC, were incredibly well liked by players, probably thanks to their instant mastery of the Like an Old Married Couple act despite (or maybe thanks to) their vastly different backgrounds and having never met each other before. Sadly, like most DLC-only characters (except Sketch from Leliana's Song), they haven't been spotted since, although the World of Thedas books reveal that they have been traveling together ever since.
Cullen, despite only having a few lines of dialogue in the original game, remains one of the most popular characters from it—mostly for the Adorkable crush he has on the female mage Player Character. His popularity was so great that he returns in a major NPC role in Dragon Age II and becomes a (romanceable) advisor to the Inquisitor in Dragon Age: Inquisition.
Bann Teagan has many admirers in the fandom, as he's one of the few unambiguously heroic characters, never steps on the player's toes, and even flirts with the female Warden.
Sandal, the simple enchanter son of the camp merchant, is fun and energetic and memetic enough that the fandom very fondly looks on him. There's also the air of mystery that surrounds him and his apparently being a supreme badass when no one's looking considering he's able to wipe out hordes of darkspawn, Templars and abominations.
Fellow dwarf Dagna is the rare dwarf that decided to study magic and is downright Adorkable. She proved to be popular enough to be recruitable in Inquisition as an Arcanist.
The (very) minor character Lilley from the "Inside Job" sidequest (who has about 10 lines of dialogue and gets killed off halfway through the quest) has somehow endeared herself to a lot of fans, some of whom go as far as to suggest she should have been made a permanent party member. This is probably due in largest part to a surprisingly powerful performance from Alix Wilton Regannote who went on to voice the female Inquisitor in the next installment, though her being a good-looking Action Girl likely helped a lot, too. There's a possibility that Lilley was supposed to have played a much bigger role in the cutCoterie questline.
Athenril seems to have attracted a similar following, probably for similar reasons, and also the fact that she's one of the elves people will hold up as a sign that the new elf design is good.
Ketojan. He really gave the feeling that he was going to be a party member, which meant his fate left a lot of players very disappointed.
Scout Harding is beloved by the fandom for being one of the few dwarves in the game, and being a badass archer with a dose of Deadpan Snarker, subtle Ship Tease, pleasant freckled look, and overall light-hearted demeanor. The fact that she isn't romance-able made her more popular, with requests to have her Promoted to Playable rather common. While a full promotion doesn't happen, the devs acknowledged her popularity by giving Harding almost as many new lines in the first DLC expansion, Jaws of Hakkon, as she had in the entire base game.
Minor character Ser Delrin Barris, a supporting character should the Inquisitor side with the Templars, has a rather large following. The fans enjoy his Knight in Shining Armor persona, his Go Through Me dare against corrupted Knight-Captain Denam, his The Men First attitude regarding his fellow Templars and his Humble Hero reaction when you compliment him. Many fans wanted him to be a companion, a romance, or at least able to speak to you outside cutscenes. Even those who side with the mages often wish they could recruit Barris.
Movran the Under, the guy who throws goats at you. He only appears for one judgement, but makes himself memorable with his affable Large Ham demeanor, throwing a goat at Skyhold to protest the death of his son (who he admits had it coming; the guy was supposed to fight the Tevinter, but instead picked a fight with the Inquisitor), and the fact that your judgement can be to exile him and his clan... to Tevinter. With "all the weapons they can carry." It's one of the few decisions that all your companions at least slightly approve of.
Coterie questline.