The Hero of Ferelden is my favorite character. Followed by the Inquisitor. And probably followed by Hawke just because she can be observed more than all of the other BioWare-made characters.
Out of all the BioWare characters other than those made by the player, I also find myself at a bit of a loss in choosing only one. There are so many of them and each has their own strengths and weaknesses. Leliana shined in Origins, but was terrible in Inquisition. Morrigan was decent through most of Origins, but hurt her own case at the end by leaving. She made up for it somewhat in Inquisition. Anora was a character I really liked in Origins, but I think BioWare underused her and made a few missteps with her that lowered her character in following installments. Characters like Sten and Oghren had some high-quality parts, but they are limited because their parts are too short or their conversations are limited.
Then there are various short-lived characters like Daveth and Ser Jory. Their roles end too quickly. This is somewhat mitigated if you use the exploit to keep them in the party through the whole game, or know their console command. Mhairi is another one with potential to be a good character, but she died too soon. Same with most origin story party companions. Their roles end before you can really explore their personas.
If I had to pick, I think it would be down to characters like Bethany, or Nathaniel, or Cassandra. Characters who either showed development over the whole game, have different outcomes affected by the player, or have their stories carried onward by the events of the series. Nathaniel and Bethany can die early, but this is dependent upon player choice. Both their fates can be decided by the player. They can both be made Grey Warden or not (though with Nathaniel it's required to see more of his story). Cassandra can't die, but she can have her fate decided in a few different ways. Restore the Seekers or not, be Divine or not. Serve the Divine or not. Each one can also have a deep connection with the protagonist. My Hero is a Cousland, and he and Nathaniel went from being at odds with each other to being brothers in arms. My Hawke is a rogue who did her best to protect Bethany, who is literally a sister to her. My Inquisitor is a male human mage who romanced Cassandra, and their connection is obvious.
So the short list is Bethany, Nathaniel, Cassandra, Leliana, Morrigan, Alistair, Oghren, and... I suppose Cullen too just because he's in all three games and can have a decent connection to all three protagonists.
Alistair, I find quite whiney, so he's not my favorite, though he has some potential and some good qualities. He's much more likeable if he hates my Warden, ironically.
Cullen has interaction with all the protagonists, and he offers a unique perspective on the events of each game, to varying degrees. In Origins he was mostly irrelevant. In DA2 he was relevant and made multiple appearances for various quests. In Inquisition he is very relevant, and has much more development in the recent game. So he gets a spot on the list, and he's better than Alistair simply because Alistair had too many faults.
Oghren is a great character, but his involvement in the story just isn't that strong. He can be recruited first or last, and with me it's usually last because the Circle and Redcliffe seem more important than Orzammar. And in Awakening he just isn't all that useful as a party member, nor as a character that relates to the story. Sigrun usually outshines him regarding dwarven matters. Plus, my Hero is a warrior, which means he's a better warrior type than Oghren. But Oghren is funny, so he still ranks pretty high on the list.
Leliana was a sweet and loyal friend in DA:O. She had few flaws, and actively sought to better herself. She strove for what was right, and she respected life. In Inquisition she was the opposite. She was incompetent and coldhearted. Her faith was broken because of the death of one person in particular. A person with plenty of her own faults, and who wasn't that important in the grand scheme of things. Inquisition really took a toll on Leliana's character. Even if she is inspired to do right again, her ideas for the Chantry are downright insane. Morrigan barely overtakes Leliana in the list because of how poorly Leliana conducted herself in Inquisition.
Morrigan is a mixed bag, but everything about her is interesting at least. She shows a wide range of emotions and motives. She is capable of being a friend, and she is capable of turning her back on one. She was irresponsible in DA:O but showed dedication to the Inquisition in DA:I. I like to think this is partly due to some regret over leaving toward the end of DA:O. She is flawed and makes poor choices, but she can be sympathized with. But she is also skilled in certain forms of magic, and has a passionate interest in preserving beneficial magic. I also like her interactions with Leliana, though there should have been more.
Nathaniel went from scoundrel to brother all within the relatively short span of Awakening. His story was heavily tied to the Cousland's. He's a character who can walk the line between shady and righteous. He has skills, intelligence, and class. He has good interaction with some of the companions, especially Velanna. He was also useful in picking locks and disarming traps. This was also right when rogues were given more talents and specializations.
Bethany is a highly interesting character. She is a sweet sister to Hawke, and she is useful. She can break Idunna's bloodmagic, her fate is heavily tied to the Mage/Templar conflict (as opposed to Carver who must CHOOSE to become a templar), and she has her own fair share of tragedy, which makes her more relatable to me, the player, and also gives her commonality with Hawke. Also, despite being taken away for most of the game, she returns at certain points, and her returns make more sense than Carver's. Kidnapped from the chaotic Circle Tower makes more sense than being kidnapped from the rigid templar order, in my mind. Bethany has a lot of personality, and I like the exchange between her and sister Hawke in Legacy when Hawke cheers her up by doing Carver's jig, and Bethany laughs and remembers the good times. Even as a Grey Warden she remains interesting. The "Sunshine" is dimmed, and her personality seems more cynical, but she also seems sharper and more world-wise. This could also make her more relevant in the future if Weisshaupt is ever explored.
Finally, there is Cassandra. She is highly relevant to the plot of Inquisition, she played a storytelling role in DA2, at least, and she's even had her own movie. She has a strong history and backstory. She has a ton of skills, both in combat and in fighting magic. Her personality is pretty dynamic. She is hardened and guarded most of the time, but she also has a softer side, which is most apparent if romanced. She can be Divine, serve the Divine, or rebuild the Seekers, so there is plenty of opportunity for her in future installments, even if it's only to help the (ex)Inquisitor stop Solas. The only real downside to Cassandra is that she seems a bit indecisive or hypocritical at times. She'll say one thing, and then disagree with the decision to heed that advice. Like with the Templar Order or the Well of Sorrows. But even if she disagrees with your major choices in Inquisition, she can still admire the Inquisitor for being able to make the tough choices. So I suppose my favorite 100% BioWare-made character will have to be Cassandra.