There are a few theories.

1 . Ned Stark is his father

• Ashara Dayne is his mother

• An unknown woman is his mother

2 . Brandon Stark is his father

• Ashara Dayne is his mother

• An unknown woman is his mother

3 . Lyanna Stark is his mother

• Rhaegar Targaryen is his father

• Arthur Dayne is his father

• An unknown man is his father

Only one can be true. There is doubt for all of them.

Jon Snow's Mother was a woman of course, most of them usually are Ive found.

Follow-up: was Ned Stark really his father?

I believe from inference of the primary sources that his mother was the redheaded whore at the local brothel. There are doubters within those who still subscribe to the concept of genetic memory because if he was the child of a whore then it is doubtful he would know as little as he did.