There is more than one kind of strong female character. It would not do justice to them to attempt to categorize them.
That said, there is one strong female character I find to be cheap and cliché: female characters who behave like men, especially stereotypically stoic men. I’m not saying such that’s wrong, or that such women do not exist in real life. I’m not even saying it makes bad fiction. However, I do feel like it makes for a poor role model. It’s bad enough that men are expected to force down their feelings to do what is needed. It seems to me like a bad idea to tell women they need to do that to be strong. Women have strengths of their own, including their ability navigate feelings, and women can strong without being masculine.
Anyway, I wanted to list some strong female characters in books I’ve read.
Honor Harrington (David Weber) is strong because she does what is right, even when it’s difficult. She’s rather unfeminine in general, and much of what is written about her is about coming to terms with being a woman. David Weber’s characters and storylines are a little simplistic, and while Honor Harrington is the most richly developed character in the series, there’s not much subtlety.
Cardenia Wu-Patrick / Grayland II (John Scalzi, The Interdependency) knows a little about being a leader from her father, and she knows that her people need her. Her femininity is not suppressed. That said, she is a strong person more than specifically a strong female.
Murderbot (Martha Wells, The Murderbot Diaries) is a sexless cyborg, and probably shouldn’t be in this list. However, most of the women in my science-fiction book club admitted that they tended to think of it as female. Take from what what you will. Murderbot is strong because it is designed to be, despite some debilitating anxieties. Over time, we also see that it is strong because it needs to be for the humans it cares about.
Winter Ihernglass (Django Wexler, The Shadow Campaigns) learns strength through experience in battle, and of necessity as a woman masquerading as a man in the military. To a large extent, this is a story of a woman behaving like a man, but there is justification in this case, because the story explores gender identity.
Cordelia Naismith (Lois McMaster Bujold, The Vorkosigan Saga) has a strength that is revealed gradually. Initially, she is the captain of a science vessel, a peaceful mission from a planet at peace, working where no trouble is expected. A few books later, she is the wife of an important ruling family, and she is literally collecting the head of her enemy, not so much because of the politics (which are significant), but to protect the life of her unborn child. She is a strong mother, a strong leader, and a strong woman. There is not a hint of stoicism or behaving like a man, and I think it is telling that Cordelia Naismith is written by a woman.
The Aiji-dowager Ilisidi (C. J. Cherryh, Foreigner) is not the protagonist of this series, but she is a certainly a strong female character. She is outwardly strong, presenting a position of ferocity and indomitable leadership. She does this simply because it has always been the key factor in her success and the survival of her house. Her inner thoughts are never revealed to us, the reader, but there are hints that she has normal doubts and fear that she masters.
Essun (N. K. Jemisin, The Broken Earth) is another character whose strength is derived from her need to protect her child. She does possess what is essentially a superhuman power, and this does give her some advantages, it is also a cause for fear and self-loathing. We see her for some time before she is a mother, and I would have to say that she is not strong then. She is somewhat stoic, but this is clearly about her state of mind, and she is definitely not taking on a masculine role. Essun is a complex character, and she is written by a woman.
Peri Reed (Kim Harrison) seems like she is supposed to be a strong character, and yet she’s not. She is a skilled fighter, and she can rewind time. She is backed by strong resources and other skilled fighters. However, she really has little or no strength of character. She is kind of a spoiled, selfish brat. I suspect this was intended to be an intentional character flaw for her to overcome, but it made her character unlikable and the novels unsatisfying.
Breq / Justice of Toren (Ann Leckie, Imperial Radch) is a sexless AI trapped in a woman’s body. She is the picture of stoicism, behaving like a computer. While she is technically a woman, in many ways she is not, and I wouldn’t call her a role model for a strong female character. She experiences feelings at times, but when she does, they are far off and foreign. She experiences even less of sexuality. This character is only one of the ways by which Ann Leckie has us thinking about gender. As for strength, she pursues her goal with the doggedness of a computer, and an angry computer at that. Her goal is revenge, inspired by an AI’s sense of love and tragedy.
Rachel Morgan (Kim Harrison, The Hollows) is an excellent example of a strong female character. She is very real, with real fears, anxieties, and doubts. She is strong in a variety of ways, and yet she mostly does not think of herself as a particular strong person. Perhaps more important to this discussion, many of the strengths she demonstrates are real, actual strengths that women possess. She has and nurtures strong bonds with her friends, and those friends add to her strength. Her feelings and her empathy are turned to her advantage repeatedly. She is far from stoic, and she is definitely not acting like a man. Rachel Morgan is written by a woman, who gifts her with a very real woman’s inner voice.
Andrea Sachs (Lauren Wiesberger, The Devil Wears Prada) ends up a strong character (in the film). She learns competence and confidence, and at the end of the film, she walks away a strong female character.