Rhetta

Strange and unappealing.

It's okay, but why not just Retta? Well, because we're extra, of course!

Not a horrible feminization but it’s a bit strange.

I would not use Rhetta (except for, perhaps, a literary character or pet), personally, but I think that it is a set aside, overlooked, underestimated gem. Depending on how people choose to pronounce it (whether the "t"s are sharp and precise, or diluted to sound like "d"s - though you could still use it, either way), Red could be a fantastic nickname (I adore androgynous nicknames - especially ones that lean a little closer to the masculine side). The more I think about Rhetta, the more I like it. It is a bit like Rita (which I also like), and vastly better than Ruby.
Hettie, Hattie, Rhea, or Ricky / Rikki could also be possible nicknames. (Along with Red, of course.)

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Rhett, male, is rather winsome and interesting, too.

Sounds like red-uh. Rhett and Rhetta look cool written, but when they're said, neither of the two sound very appealing.

This name is so hopelessly old-fashioned.

Has anyone actually ever named their daughter this ugly name?