It's no secret by now that Owain, Inigo and Selena have returned in Fire Emblem Fates. If you didn't know that, that's what the spoiler was for. Anyways, back in Awakening, Lissa, Owain/Odin's mother had the option of marrying Henry, an insane and child-like dark mage. In Fire Emblem Awakening, Owain's default class was a myrmidon, a sword fighter. However, we can assume that his father, Henry, taught him a few curses and dark magic, causing Owain to become fascinated with it. In Fates, his default class is a dark mage as well.

Now, the strongest piece of evidence I have is through photos.

I divert you to the hair Owain has in the portrait with Henry's name on it, signifying this is Owain's hair color with Henry as his father. Now I encourage you to look here.

His hair looks white. Maybe it's light yellow, and there's a giant hole in my theory. I don't know. What do you guys think?

5166108 Odin's canon father is whoever Lissa married in your first Awakening playthough (As confirmed by Hidden Truths.)

Anakos changed their hair color in Hidden Truth's, so its unlikely they would still have the same hair color as their father. Instead, they all have the hair color of their mothers, just like in their original official art. So His hair is the same color as Lissa.

That being said, I do think that Henry is the most likely father of Odin/Owain due to the many similarities they have. Odin and his daughter's supports reference Henry's supports more then anyone else if at all, implying a strong connection. In his supports with Selena/Severa, he helps his sleep with a spell, just like the one Henry uses to help Lissa sleep, sans a bird sacrifice. Ophelia preforms a body switching spell with Soleil in a manner almost exactly like Henry's support with Sumia. Once is a coincidence, twice is just teasing, especially since the second involves a reference to a support between two characters they aren't strongly related to.

I'll accept this headcannon only with the caveat that Donnel is Inigo's father.

Awakening somewhat implies Lissa/Lon'qu more strongly, though.