There are many problems with the idea of the strong female character, although the trope started out in the right place. To try to make amends for the sexist and problematic "damsel in distress," progressive writers set out to write female characters characterized by a rejection of typical female gender roles. However, the perception of what makes a strong female is still grounded in the very thing the trope aimed to escape--the idea that masculinity is superior.
The most common "strong woman" trope shows a battle-hardened warrior, or a spunky thief who rejects femininity. All of these have to do with a masculine ideal. In order to be categorized as strong, the female characters must reject femininity and display traditionally masculine traits such as aggression, violence, and an unwillingness to show emotion. It's good to have characters that reject gender stereotypes, but when it's the only type of strong female character seen, it still perpetuates the myth that strength is inherently masculine.
Mere physical prowess does not make a strong female character. Too often, the strength of female characters is solely defined by their ability to beat up villains or shoot a bow and arrow. They are defined by their "toughness" and the way that they're "not like other girls," as if femininity is a weakness.
What makes for a strong female character, then, if not physical strength? Well, here's the inherent problem--there is no set trait, because women are human. We need to stop focusing on writing "strong female character" and start writing humans. They may be feminine, or masculine, or somewhere in between. They may hide their emotions or display them in full force. They may be leaders or they may be followers. None of these traits are stronger or weaker than any of the others.
Take Cath from the book Fangirl by Rainbow Rowell. She's insecure, suffers from extreme anxiety, and deals with things like her father's psychotic breakdown and losing touch with her twin sister. Her strength is not in any masculine physicality or in spunkiness, but in her ability to rise above her life problems and accept her emotions. Cath is strong because she accepts her emotions, not because she hides them. Strength doesn't lie in the ability to brush off emotions or grief, but in the ability to accept them and be unafraid to display them. Too often people are told that to be strong you can't break down and cry, yet some of the strongest characters are the ones who aren't afraid to use their emotions to grow as people. There's a quiet and unshakable strength in the realization that you need help working through, for example, a mental illness, and rising to the occasion to ask for it and lay out your emotions for others to see.
Offred from The Handmaid's Tale by Margaret Atwood isn't a raging warrior, and we aren't even sure if she manages to escape her prison at all. She lives in a society where fertile women are forced into sexual slavery and aren't allowed to read. She is in a position of absolute powerlessness, completely controlled by the Commander, the man she is enslaved to. And yet her strength still shows through, in her subtle acts of resistance. One of the most poignant small details of the book is when she mentions stealing flower petals from the vases because it is something the Commander will never know about, a way to evade and defy him. Her ability to keep fighting even when in a position of absolute power, and her ability to eventually take action for the resistance group Mayday and defy Gilead's rules by making love to a man purely for her own emotional needs rather than to reproduce, make her an amazing character.
Or, for our last example, examine Elphaba from Wicked. Here is an extremely flawed, prickly, and eventually delusional character who somehow manages to be one of the most memorable characters of all time. Instead of being the one-dimensional Wicked Witch we saw in The Wizard of Oz, we see someone deeply flawed and yet extremely interesting. The mix of dark and light within her is one of the central themes of Wicked. She's mean and not afraid to get revenge, even on young children, yet she's passionate about the rights of talking Animals (even going so far as to save the young Cowardly Lion from experimentation), and has a deep love for others.
What ties all of these very different characters together and makes them all distinctive, strong female characters? The answer is their humanity. The characters are first people, and then women. They are complex, with terrible flaws and incredible strengths. They have deep desires, fears, and goals. They change and develop, they dive into the deepest desires of humanity in their search for connection, love, and control over their fate. They are also women with their own agency, even when the world is attempting to crush them under strict regimes or under social expectations. They power through and find their own personal goals, be they traditionally feminine or masculine ones.
In fact, screw those labels entirely. Femininity and masculinity are too ill-defined anyway. What makes someone womanly or manly? The characters outlined above are all feminine in different ways. They search for their own definition of being a woman, and by extension, a human. Women are human, which means they should be just as unique and complex as men. The spunky heroine, the damsel in distress, and the masculine warrior are only ways to simplify women into select types. But you can't define a human with only types. These types can be used as a starting off point, but if you truly want a strong female character you're going to have to put in the work of making her a unique human being with all the messy flaws, emotions, and goals that come with that (I'm saying human because it's the most typical species for a character, but by all means feel free to apply this advice to your alien, rabbit, or sentient plant characters as well).
To get you started thinking about ways to develop your female character more, here are some questions to consider: What are the gender roles in the society she lives in? Is she expected to adhere to traditional femininity, or are women placed into some other role? Or is the society better off than ours, one in which people of all genders are treated equally? To really get in touch with invisible social expectations, think about what the society defines as an ideal woman, meaning the culmination of everything that a woman is expected to be. For example, throughout human history, the ideal woman was:
--eager to please
Think of factors such as if a hierarchy exists between women. If women are oppressed in any way, there's going to be a social hierarchy, because oppressed groups have historically kept their own members down due to the desire to become favorable in the eyes of the powerful. How does the society view sex? Marriage? Menstruation? Queerness? Does the age of a woman play a role in how she is treated?
Once you've established whatever society your character lives in's view of women, you need to define your character's relationship with that role. Does she strive to be the perfect woman, does she consciously rebel against it, or does she just ignore it? Any way she does it, she is going to be aware of the expectations placed upon her. A woman who is assertive towards men and speaks up for herself is aware that she will often be seen as bossy, raging, or melodramatic. Every day of her life, she will fight against that idea and deal with men who view her as a rabid feminist or hysterical whiner. In your story, your character should be aware of whatever roles exist, whether she's aware of them consciously or subconsciously. She might compare herself to other women, become hostile towards women who adhere to the traditional norm, or willingly ignore whatever gender roles exist. This will all depend on the character, but it's important to define the ideal woman and your character's relationship to her in order to write how she interacts with society.
How does your character view sexuality? Does she see it as a shameful thing, or does she embrace it? Does she treat people of all genders the same? Even if it's subconscious, there's likely to be a subtle shift in the power dynamic when she is interacting with men. How would she react to unwanted sexual advances? Would she call the person out on it, or would she remain silent? What is her greatest fear? What is her greatest dream? What kinds of people does she look up to? What kind of positive role models did she have throughout her life? What are her strengths? What are her weaknesses? Is she happy with her life, or does she feel chained down? What does she think of her body? Does she feel confident in it, or does she want to change it? Is she the kind of person to want to display her body, does she prefer to hide it, or does she not care one way or the other? Is she comfortable showing emotion, or does she suppress it? If she's faced with a problem she can't solve on her own, will she seek out help? From whom? If she's having a breakdown, is she likely to reach out to others? Does she struggle with any mental illnesses, such as anxiety or depression (if so, please, please, please do your research, and don't add it in nonchalantly or bring it up only when it's convenient).
The answers to these questions might not play any sort of role in your story, but it's crucial that you know the answers just the same, because they are key to understanding your character as a woman, and even more importantly as a person. Once your character is a 3-dimensional human, with flaws and strengths like everyone else, you've got a strong female character. If you have any comments or advice, be sure to share.