Are there specific characteristics that one should strive for when trying to develop a strong female character? Is it truly dependent on the genre of the book, or are there certain elements that carry over across all genres?
WARNING: This answer contains numerous links to TV Tropes, an irreverent taxonomy of common tropes in film and fiction. TV Tropes is highly addictive, wasting hours of "just checking the definition of one more term." You have been duly warned.
Obviously, "strong female character" can cover a lot of territory! But here's what I see as the general considerations.
The idea of "strong female characters" is reactionary in nature. It's reacting to the sense that these characters are something unusual to encounter in fiction; a rarity; something noteworthy. Nobody said "hey, there aren't enough blind seers in my fiction!", but they did say, "hey, how come my fiction doesn't have as many awesome, capable women as it has awesome, capable men?"
So to understand what's being asked of a strong female character, we need to understand the trope being subverted. (I'm moving to gross generalizations here, and far less common today than they used to be.) Commonly, female characters serve primarily as romantic interests, as damsels in distress, and/or as the Token Chick. These characters are not strong, in the following sense: they seem to be defined entirely in relation to central male characters. They exist to enable his story; they don't have a story of their own. They are not the star. Commonly, they simply aren't as capable and talented as the male lead - or are Faux Action Girls, ostensibly as talented as the male, but in practice never doing anything heroic.
So the farther we manage to move from the supporting, subservient, and diminishing role that female characters are commonly assigned, the closer we are to a "strong female character." Here are a few varieties:
- Action Girl: This basically subverts the trope by inverting it - making a female character awesome, strong, a skilled warrior, a deadly shot. You take the stereotypical action hero; swap genders, and bam, you have produced a strong female character. Well, by some definitions. See enough of these, and they get very simplistic, and not a whole lot deeper or more representative of women in general than the original trope was. OTOH, when done well, it gets around the bad trope's biggest problems: the female character has agency and independence, and is central to the plot rather than supporting other people's stories.
- Gender Equality: In much the same way, any "typically male" character archetype can be fitted to a female. So the trick is not to settle on male character just because they (allegedly...) come to mind first. Rather, populate your fiction with just as diverse a cast of female characters as your cast of male characters. Likewise, portray women in all the situations, careers and positions they can plausibly get to - not only the ones that "best fit a female character."
- Uniquely Female: Probably the strongest characterization is that which manages to portray strong female characters, but without feeling these are mirror images of men. Rather, they are unique characters, individual, believable, whose personality is affected by their gender - as everyone's must be - without falling into stereotypical, belittling notions of how women's lives look.
Here's the fundamental element: to be a strong female character, your character must be capable, must have some degree of agency (i.e. goals of her own and the ability to pursue them some of the time), and basically must earn the reader's respect in her own right. She's "strong" in the sense that she can do stuff, and no less than that - in the sense that the stuff she wants to do is important.
Beyond that, she can be a sharp-tongued private sleuth or an adorable vampire hunter or the resident control freak. Character can vary infinitely; that's practically the point - that females aren't reduced to a few supporting, nurturing, handily-refrigerated roles. If she's impressive, if she's meant to gain the reader's respect and not just his tension or romantic interest or sense of accomplishment - that's a strong female character.
Note: I'm not saying, of course, that there's any expectation that all female characters in a story be "strong," or even any of them - no more than all the male characters need to be. The issue is more one of total volume - one story makes no difference; when you start noticing trends across hundreds of novels, it's a bigger deal.
A character is a character is a character is a character regardless of genre. A character is a set of behaviors (words and actions) and reactions to those behaviors by the other characters. Even the physical description of the character can give us clues to her personality. How she carries herself.
Assuming I'm interpreting what you mean by "strong" in the way I would interpret it, a woman who is confident and assured, but at the same time not in-your-face aggressive, make sure her actions are well-considered. If she has to make a decision, don't make her vacillate. Make her confidence such that other sensible, balanced people don't mind deferring to her judgment.