I know that male circumcision had some health benefits back then (or so I've read) but how did Female Genital Mutilation become a thing in African muslim countries. I am especially curious about Egypt becuase they are the most "modern" country that still does it frequently (90% of the time according to wikipedia).
The reasoning behind the rise of circumcision is not "health benefits" if you look into History, it's a religious/tribal sacrifice meant to brand and assert domination over people's sexuality.
Maimonides explained it in the Guide of the Perplexed (http://www.cirp.org/library/cultural/maimonides/)
All tribes/communities that perform female circumcision perform male circumcision as well.
I think you'd be very surprised by the thought process that's behind it, the marketing changes over time of course (health benefits, aesthetic, source of pride for belonging to this or that community, purity,...) but it can be most accurately described as a cyclical abuse.
There's an ethnologist that studied this practice and got circumcised herself at 21yo after studying in the US, "to identify with the women in her community" and she actually promotes the right to continue this ritual.
http://www.fuambaisiaahmadu.com/
She wrote her thesis on it and you can find a few excerpts on her website. For the record, in the vast majority of cases, the people who perpetrate female circumcision are women, they see it as the norm just like cut men here insist on keeping the practice going for their children. Victims of it most of the time do ot see it as harmful when they are in their home countries, it's just normal, they are culturally desensitized to it just like many people view male circumcision as normal/positive.
Note that there are different types of cutting for each sex, the most invasive one for women being infibulation (which must be undone if she gets pregnant, very dangerous otherwise), there are many other ones, some of which are done in the West in plastic surgeon offices such as labiaplasty, vaginoplasty, clitoral hood reduction or clitoropexy, they would qualify as female circumcision/genital mutilation in other contexts. The popular term for it is "designer vagina".
I can't answer your question for Egypt, some hadiths recommend female circumcision but it's not widely accepted among muslims and it's not mentioned at all in the quran (neither is male circumcision). It depends on the culture you're dealing with. That 90% rate is very surprising considering that 12% of Egyptians are christians, I'm not sure it's accurate for EGypt as a whole, the practice has been banned in this country since 2008 .
It must be noted that it's not only a muslim thing - animist and christian communities (Bondo in Sierra Leone for instance) practice it as well.
It has also been practiced in the USA in the 19th and early 20th century to "cure" lesbianism, masturbation and hysteria. Edwin Pratt and John Harvey Kellogg were the main proponents of this approach. You can look up their theory on "orificial surgery".