Some fanfictions might choose Hermione as the mother. This could create an interesting dynamic, as Hermione is Harry's best friend. It would bring a whole new set of relationship developments and challenges for Harry as a father.
There could also be original characters created just for the fanfiction to be the mother. Maybe a new witch that Harry meets during his post - Hogwarts adventures and falls in love with, leading to the birth of their child.
It could be Ginny. In the original series, they had a relationship. An unexpected turn of events could lead to her being the mother in this fanfiction.
It could be one of Harry's enemies who is trying to frame him. So they set up the situation to make it seem like Harry is the father.
One way this could work is through some sort of magical time - traveling or dimension - hopping accident. Maybe Naruto, in his own world, gets caught in a strange jutsu that sends him to the wizarding world of Harry Potter. Once there, he meets Harry Potter's mother, Lily, and they fall in love. Naruto, with his strong sense of justice and loyalty, would be a great father figure. He could teach Harry about the importance of friendship and never giving up, which are values that Naruto holds dear in his own story.
In this fanfiction, the 'Father of Monsters' might be an ancient wizard who has been in a deep slumber for centuries. He was awakened by some disturbance in the wizarding world and, feeling angry about the changes that have occurred, decides to create monsters to bring back what he believes was the 'true' order of the wizarding world. His knowledge of old magic gives him the ability to create extremely powerful and dangerous monsters that no one has ever seen before.
Another possibility could be Draco Malfoy. Think about it. In some fanfictions, they might rewrite the relationships completely. Draco has his own complex character arc, and perhaps in this 'harry potter is not the father' scenario, he ends up being the father due to some unforeseen circumstances, like a misfired spell that alters the course of events related to parentage.
It could be Sirius Black. Maybe there was some misinformation that made Harry believe Sirius was his father for a while, but then it turns out he isn't. After all, Sirius was very close to Harry and had a father - like relationship with him in some ways.
Annabeth could be the mother. Since she and Percy have a strong relationship in the original series, it's very likely that in a fanfiction they would have a child together. Their combined intelligence and Percy's powers would make for an interesting genetic combination for their child.
If there is a mother or father in the fanfiction, it could be another angel. Maybe an angel that Castiel had a past relationship with in the celestial realm. This would add a layer of drama as they might have different ideas about how to raise the baby or what the baby's purpose is.
It could be a completely new character created for the fanfiction. Maybe a powerful witch who met Sirius during his time on the run. She could have been helping him and they fell in love, and then had a kid.
The Ministry of Magic could turn into an antagonist. If Harry's new cold and strong ways are seen as a threat or too radical, they might try to stop him. This could lead to a conflict between Harry and the wizarding government.