The main thing that makes me want to write genderswap is the feminist/gender studies/queer studies angle: how can you use it to interrogate gender roles and gender politics either in a series itself, or in its fandom? And you can do different things with that, I think, depending on what kind of genderswap you go for. Choosing the "X wakes up one morning/comes to from a sudden blackout/&c. and has switched gender as a result of a magic potion/curse/rift in the fabric of spacetime/botched channeling job (I guess the easiest route in this fandom)" route turns the genderswap into an explicit foil for revealing other characters' (and the character's own) biases (because you can structure a contrast between the pre-swap and post-swap treatment of the character, and isolate gender presentation as the factor influencing the change), and can be used to produce a "revelatory" kind of effect: "My eyes have been opened to how much different my experience of the world would be if I were a different gender." Also, because usually this kind of genderswap is at heart "Character X in a different body", with the physiological changes having maybe a little impact on their personality, you can also produce a similar revelation in the reader, or at least get them to ask themselves: would my attitude towards this character have been any different if (assuming all else is equal) they had been a different gender? (Klavier's an interesting one to think about this with)
If you go for the "All else - circumstances of birth, genetic makeup, upbringing, endpoint in canon - being equal, Character X was born and has always been a different gender" route, then you've got different options. You can switch a whole cast, and see if that makes any implicit biases or absences more transparent - or explore what kind of world the series would have to be set in to produce the gender-reversed makeup of the cast. You can take a very focused look at one specific character, and consider sexual politics through the lens of their identity: take apart how central their gender identity is to their canon personality and characterisation, think about what about them would have been the same and what would have been different if they had been born a different gender, and through that take apart just what part gender plays in shaping people's identities. It quite often, too, exposes how the roles of characters in canon are implicitly gendered - would the exact role of Edgeworth ever have been written for a woman, for example? (it's probably fairly obvious that this second type is the type I prefer to read and write - but I am willing to be converted to the first one!)
The other reason that's led to me writing the particular genderswap I am is purely porn-related. I badly wanted to write a very specific kind of sleazy femslash smut - the sort of sex you'd get between Klavier and Daryan, or Matt and Juan (except maybe with a little less hating each other) - but I couldn't find an f/f pairing in this fandom (or any fandom I'm in, really :/) that it seemed IC for (suggestions, if anyone's got any, would be welcomed!). So, fixation with genderswapping Klavier (more on that in a minute) + desire for Klavier/Daryan-esque femslash = fem!Klavier/fem!Daryan. Success. From what I've seen on the meme recently, this rationale seems quite common - using genderswap as a way of preseving a particularly appealing pairing dynamic while being able to enjoy a different (more appealing?) configuration of bits :D
So I do get genderswap, I do. I think the thing I don't always get is how it works with these games particularly.
At heart, I think this is probably because the first three games, at least, don't have any seriously questionable sexual politics (compared to a lot of other media, at least ) that I'd really want to see dealt with - and in fact, there's actually a lot about the gender makeup of the cast that I think is very positive and interesting and wouldn't like to see reversed. The Fey clan in its entirety, for one thing - I wouldn't want to lose the fact that it's a female-centric dynastic plot underpinning the first three games, for example, not to mention all the different relationships represented between women as a result. Mia specifically is already a woman in the more usually male role of mentor to a young man, and I love that Maya is a female sidekick whose relationship to the male protagonist is not sexualised. Meanwhile, genderswapping Phoenix or Edgeworth (I'm not sure if my point would still stand if you swapped both) and then getting them together could be an interesting way of showing (as I actually think Takarazuka's character design for Leona does) that if either of them were a woman, their relationship starts to look like one that would be broadly perceived and understood as romantic - but for me (and ymmv) what's interesting about their relationship is the fact that it is in canon such a romantic male-male friendship. Generally speaking, furthermore, there's not really a lack of either male or female characters - and there's certainly not a lack of possible pairings/OTns, so I tend to find I can get what I want smut-wise and gender-representation-wise from what's there in canon (of course, that said, as I guess I've illustrated above, this doesn't always hold true, and I've no doubt that other writers/readers have found that canon's not giving them quite what they want to write or read in this respect). And as a result of a relatively good balance between male and female characters, the world of the games itself comes across (comparatively, at any rate - which given the state of a lot of media isn't necessarily saying much) as being relatively egalitarian, which is something that makes genderswap less compelling, in my eyes.
It's the aspects of the game that are problematic that would for me generate the most interesting genderswap. Getting Diegodot involved, for example - one of the genderswaps I've enjoyed most in this fandom was a Miego fic where Mia ended up male-bodied after a channeling went wrong, and Diego had to deal with that (with sexy results). Or Dahlia and Iris, both of whom stay just on the right side of stereotype for me - I think it could be fun to see how Dahlia's particular brand of evil reads when you detach it from the lineage of beautiful, poisonous, evil women she sometimes feels like she belongs to, or how Iris's traditionally feminine qualities read when embodied by a man. Genderswapping Franziska might work well, too, not because her character is particularly problematic in my eyes, but because a lot of the reactions of other characters to her in the games seem a little gendered to me. And there's a lot you could do with Apollo Justice, given that the gender issues of that game are different and more concerning.
In the end, I think whether or not I "get" genderswap fic fandom really does just depend on the characters. The sexual politics of the series generally I don't find troubling enough to want to write or read srs bsns feminist genderswap - individual characters, OTOH, are a different story, and I do find that for me, some characters genderswap better or more interestingly than others. miss_prince said to me when we were talking about this that she finds genderswap most interesting the closer the character is to traditional masculinity or femininity (giving Tigre and Iris as examples), and I'd agree - I'm personally looking for where genderswap would produce the biggest or most revealing difference, either in the experiences a character is likely to have because of their gender presentation, or the difference it might make to the player/fan's perception of them. This is why personally, I'm really keen on genderswapping Klavier. My wanting to write genderswap actually came out of talking to (IIRC) alienchrist and shahni about if or how fans would treat Klavier differently if he presented as female - how his charm, his attractiveness, his flirtatiousness, his high self-esteem, his love of the spotlight, his relentless cheerfulness and the points where his emotional displays are odd or awkward, his prodigious success, the (really kind of implausible) fact that he works two incredibly time-consuming jobs and is extremely successful at both of them, just generally his persona and the way he relates for people, would have been judged and received if we were talking about Klaudia, rather than Klavier, Gavin. Because it seems to me that despite the androgynous and not explicitly gendered aspects of his personality, there's a lot about Klavier that aligns with acceptable paradigms of masculinity and not with acceptable paradigms of femininity. And then I started to think about how far Klaudia Gavin might have ended up a different person as a result of different pressures or expectations - would she have become a different kind of flirtatious? Might she have had to prove herself in different ways? What might her confidence and high self-esteem have become in the face of different attitudes towards it? Would she have been a different kind of performer, and would a woman-fronted (or all-female) Gavinners have been a different kind of band? And then I couldn't stop thinking about it, and so I started writing a genderswapped Klavier/Daryan fic that one day I swear I will finish.
I'm sure, though, a good fic writer could ask similar questions of a whole range of a whole range of characters and come up with interesting answers where I'm just drawing blanks (cards on the table: I personally don't get the appeal of genderswapping Phoenix or Kristoph at all, for porn purposes or otherwise, and in fact it was the recent proliferation of fics that do that's been pushing me to think harder about the appeal of genderswap beyond what I'd want from it) - and I'd love to hear if there's any characters in particular you guys would be particularly interested in genderswapping, and why.
The final thing that makes me think sometimes is whether "why (or "how to") genderswap?" is necessarily the most interesting question to ask about it. Whether it's worth instead picking apart the concept itself from a gender studies perspective, and whether it relies on particular notions of gender (throughout this whole thing, for example, I've presupposed a gender binary of one form or another) - or asking how (if at all) it relates to trans issues, for example. Or if it's best approached just as a literary device, and one that relies on a particular set of assumptions and conventions to achieve a particular set of effects. It's not an angle as yet I've thought much about, though, idk about anybody else?