One idea could be that they travel back to the time when the Targaryen dynasty was at its peak. Jon could see Daenerys in her element as a Targaryen princess. They might try to change some of the events that led to the downfall of the Targaryens, like preventing the Mad King's actions. But this could have unforeseen consequences for the future they came from.
In fanfiction, one way it could be possible is through some sort of magical spell or ancient artifact. Maybe there's a long - forgotten Targaryen relic that has the power to manipulate time. When their child discovers this relic, it accidentally sends them on a time - traveling journey.
There's 'Time Traveling Dragons' which is an interesting take. They are sent back in time to a period where dragons were more common. Daenerys gets to see how her ancestors interacted with their dragons, and Jon is there to support her. The story has a lot of twists and turns as they try to find a way back to their own time while also changing some things in the past for the better.
In a time travel fanfiction, their son could go back to the moment when Jon and Daenerys first met. He could subtly change their initial interactions to make them trust each other more quickly. This could lead to a stronger alliance from the start. He could also travel to the times when they faced their biggest challenges individually and offer solutions. For example, when Daenerys was dealing with the slavers in Slaver's Bay, he could give her ideas on how to end slavery more effectively and gain more loyal followers, which would have put her in a better position when she reached Westeros.
It could also develop more complicatedly. Suppose they travel to a future where one of them has made a big mistake in the fight against the Night King. This could create mistrust or a power struggle between them. Maybe Daenerys blames Jon for something that went wrong in the future battles, and they have to work through that while still trying to deal with the Night King threat.
It would greatly affect Jon. If their son travels back to the time when Jon was still at the Wall, he could reveal to Jon his true identity earlier and help him accept his Targaryen heritage more smoothly. This could change Jon's actions and decisions. For Daenerys, the son could give her more confidence in her claim to the Iron Throne by showing her the future support she could have had.
There could be. Fanfiction is a vast world where fans can take the story in any direction they like. Since Jon and Daenerys were significant characters in the series, it's not uncommon for fans to write about them. Regarding a story where Daenerys is pregnant, it might be a way for fans to explore their relationship further, perhaps imagining a different future for them. You could also try looking on Wattpad in addition to AO3 for such fanfictions.
Tyrion Lannister could be involved. He was close to Daenerys and had a relationship with Jon as well. He might be the one to help protect the baby or offer his wisdom in raising the child. Another character could be Ser Davos. He was loyal to Jon and would surely be interested in safeguarding the future of Jon's heir.
Well, a simple idea could be that the pregnancy is a secret at first. Only a few close allies know. Jon and Daenerys have to be very careful to keep it that way. They might have some close calls where the secret almost gets out, adding a lot of tension to the story.
Common themes include destiny. In many 'jon and daenerys fanfiction', they are seen as fated to be together, whether it's to fulfill some ancient prophecy or just to bring about a new era in Westeros. Family also plays a role. Their respective family legacies, the Starks and the Targaryens, are often explored. For example, how their union might heal the rifts between the two families. And then there's the theme of sacrifice. They might have to sacrifice their own desires or even their lives for the greater good of the realm, which is a popular plot point in such fanfictions.