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 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.
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.
During this time, she met Rayleigh and they fell in love. Shakky retired before the God Valley incident,(still to be confirmed, but she might have joined rocks pirates) 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.
I'm Regards to his eyes, an extremely rare male Kuja infant could be born with features that we don’t see often, such as red eyes or hawk-vision! Mihawk’s eyes could also be inherited from his maternal grandpa, who we know nothing about. (or it could be just Oda's design trying to show mihawk intensity.
Mihawk being Ray’s son also explains his friendly rivalry with Shanks, who must have been Rayleigh’s swordfighting student.
So, I admit that the connection to Amazon Lily and the eye theory are a little out there, but I’m 99.999% certain that Rayleigh and Shakky are Mihawk’s parents, whether the Amazon Lily part is true or not.
I got this theory online and I think it makes sense.