I decided to finally consolidate some headcanons and theories about Mihawk’s parents.
Parts that are confirmed canon are in bold. Feel free to take the rest with a grain of salt.
I think the manga has given us enough clues that Mihawk is Shakky and Rayleigh’s son, including the fact that she retired “40 years ago” and the increasing frequency Oda is showing us young Rayleigh wearing the same paisley pattern on his coat.
The only other reason I can think of that explains the paisley pattern is that the paisley is some kind of WorldStrongestSwordsman-only pattern, only worn by the best–like the yellow jersey in the Tour de France, lol.
Mihawk looks quite a bit like Shakky and seems to have the same hobbies (drinking and reading the newspaper).
All of the women of Amazon Lily have flower names. Shakky (whose full name means “Peony” in Japanese), is Gloriosa’s daughter.
(I think the shirt Mihawk wore to Roger’s execution has peonies on it too!)
Gloriosa was once the Empress of Amazon Lily, but she got love-sickness. The person she fell in love with was Shakky’s father.
In Japanese, Hancock calls her “former-former-former Empress” (先々々代皇帝).
So the Empresses were:
Gloriosa → (Empress1 who had love sickness and died) → (Empress2 who had love sickness and died) → Hancock
Furthermore, Gloriosa says that she survived by fleeing Amazon Lily. I think that’s important, because it means that she left to be with the person she was in love with rather than wasting away pining for them.
Gloriosa and whoever she was with raised their daughter Shakuyaku on the sea. Gloriosa was a Kuja warrior back in the day, and Shakky was brought up to be a fighter and a pirate.
So Empress2 above was still alive when the Boa sisters came home, and she pardoned Gloriosa. When she died, the women of Amazon Lily made Hancock the new Empress.
Back to Shakky, she joined Rocks at some point (only half-bolded bc it’s not confirmed yet xD). During this time, she met Rayleigh and they fell in love. Shakky retired before the God Valley incident, likely because she got pregnant with Mihawk.
Gloriosa might have come to Sabaody to help her daughter with the baby, which would explain why Rayleigh knows Gloriosa and calls her by her name instead of Granny Nyon. Because they were both pirates, Rayleigh & Shakki gave their son a different family name than either of theirs.
This is also why Rayleigh and Shakky helped Hancock and her sisters on Sabaody before Gloriosa took the three girls back to Amazon Lily.