I'm not sure what I'm about to mention will be helpful -- I would think it might be most effective to work with what the mother already wants and believes, and from what you wrote she wants her daughter old-for-grade.
However generally, as regards acceleration/on-time/red-shirting for girls, here are a couple of thoughts, looking forward into the daughter's future:
(1) There's been quite a bit of research on timing of puberty vs. girls' self-esteem/body-image/susceptibility-to-dating-violence etc. And keep in mind that very frequently girls are developing by the end of fifth grade nowadays -- so this may be coming upon your daughters a LOT sooner than the friend's mother may be anticipating! (Talk to fifth grade teachers and take a look yourself at the fifth grade girls at your kid's school and how they change from the beginning to end of the year.) Generally, the research tends to show that girls are happiest when they are "in the middle to slightly late" in developing, and most unhappy/at risk if they are early developers (possibly excepting African-American girls who navigate the pubertal transition with much less drop in self-esteem).
The friend's ethnicity, weight status, family history of pubertal age, and whether or not she lives with her biological father in the house can all affect pubertal timing, so maybe this will or won't be an issue for her.
In contrast, for boys it's cool to develop early, and the most unhappy/at-risk boys are those who develop late.
So to me, this is one reason why red-shirting boys makes more sense for boys than for girls: for boys, early development is all good, but for girls, the greater mental maturity and physical development for sports coming with old-for-grade has to be balanced against the potential social and psychological costs of being an "early developer" among grade peers.
Perhaps the mother might be open to considering this trade-off: even if keeping her daughter old-for-grade is working great right now, will her daughter have to pay for it with extra stress later as an "early developer", maybe even at age ten in fourth grade? If skipping to match the daughter with age-mates is feasible now, it may avoid some grief later.
(2) A common belief is that kids will all end up pretty much the same place anyhow, that doing something one year sooner or not doesn't matter, so why "rush" them with a grade-skip? Counteracting that belief are studies such as Park, Lubinski, and Benbow "When Less is More: Effects of Grade Skipping on Adult STEM Productivity Among Mathematically Precocious Adolescents", which indicate that acceleration for the very gifted (more so for boys than girls) doesn't just put kids inline to achieve the same milestones a year earlier as adults, but rather puts kids inline to achieve significantly more overall as adults -- that the "small" acceleration of a grade-skip as kids sets them on a much higher achieving track in the long-term. A grade-skip is a very individual decision. But it may be helpful to be aware that there are indications that grade-skipping, when appropriate, can indeed result not merely in moving forward at the regular speed just one year earlier, but rather can result in achieving significantly more throughout adult life.
