While Jon Snow (Kit Harington) was leading his Suicide Squad to capture a Wight in the sixth episode of Game of Thrones season 7, "Beyond the Wall," Daenerys Targaryen (Emilia Clarke) and her Hand of the Queen, Tyrion Lannister (Peter Dinklage), had a heated discussion about succession. In the event that Daenerys does win her war with Cersei Lannister (Lena Headey) and captures the Iron Throne, the pressing matter of who would become ruler of the Seven Kingdoms after her is something that causes both Daenerys and Tyrion great consternation. After all, in season 1, a witch cursed Daenerys with not being able to have children; she has accepted that her three dragons are the only "children" she will ever have.
The romantic relationship between Daenerys and Jon could solve this sticky problem. Jon finally bent the knee to Daenerys and accepted her as his Queen; Daenerys in turn accepted Jon into her bed in the season 7 finale "The Dragon and the Wolf." Yet this inevitable Jon and Daenerys coupling could complicate matters even further. Neither is currently aware that Jon is actually a Targaryen and Daenerys' nephew. He is the son of Rhaegar Targaryen and Lyanna Stark, and a true blooded Targaryen who has a legitimate claim to the Iron Throne.
Even when Jon and Daenerys eventually discover Jon's true heritage, it doesn't necessarily mean there's a problem. The Targaryens have a legacy of marrying family members to keep their bloodline pure. However, the eventual reveal of Jon's heritage could open up a bevy of problems for Jon, Daenerys, and for the entire country of Westeros.
IS JON SNOW THE RIGHTFUL HEIR?
This is the biggest question that has to be addressed: Does Jon's claim to the Iron Throne supersede Daenerys' claim? As Gilly (Hannah Murray) discovered while reading the meticulous records kept by High Septon Maynard at the Citadel - which was confirmed by Bran Stark (Isaac Hemstead-Wright) traveling back in time as the Three Eyed Raven to witness it - Rhaegar Targaryen secretly married Lyanna Stark in Dorne. Furthermore, Rhaegar's previous marriage to Elia Martell was annulled; therefore the legal claim of his children with Elia, Rhaenys and Aegon, was rendered null and void. As the Prince of Dragonstone and first born son of the Mad King Aerys, Rhaegar was heir to the Iron Throne. With his annulment to Elia and legal marriage to Lyanna, this means that his son with Lyanna, Jon Snow - who was named Aegon Targaryen at birth - becomes the heir to the Iron Throne, especially since every other Targaryen is dead. Every Targaryen... except Daenerys.
Daenerys is the younger sister of Rhaegar and only living child of the Mad King Aerys. Her claim to the Iron Throne rests entirely upon her family name and her family's history as the founders and rulers of the Seven Kingdoms. Plus, ruling the Seven Kingdoms has been Daenerys' one goal and lifelong ambition. From everything we've seen throughout the 7 seasons of Game of Thrones thus far, if one person has proven she actually deserves the Iron Throne and would be a good ruler, it's Daenerys.
Though the Targaryens were deposed by Robert's Rebellion and the ascension of Robert Baratheon (Mark Addy) as King, it can be argued that since neither King Joffrey (Jack Gleeson) nor King Tommen (Dean-Charles Chapman) were actually Robert's sons - they are the product of incest between Queen Cersei and her twin brother Jaime Lannister (Nikolaj Coster-Waldau) - Westeros hasn't had a true King since Robert died. And yet, even with the rampant suspicions about Joffrey and Tommen's true parentage - and now the outright flagrant flaunting by the reigning Queen Cersei of her incestuous relationship with Jaime - the systems in place in Westeros (controlled the Lannisters) continue to support the Baratheon-Lannister claim to the Iron Throne.
Hence, Daenerys doesn't have legal right to automatically claim the Iron Throne, in spite of her family name. She must win it from Cersei through war. This war is overall going in her favor, and should Daenerys ultimately win, the Iron Throne is hers' - until Jon's heritage and claim come to light. Then the issue of whether Jon has more rights as Rhaegar's son and heir versus Daenerys' rights as Rhaegar's sister will have to be settled. There are a couple of ways this can be sorted out, without getting the Maesters of the Citadel and a lot of legal wrangling involved.