Although Jon Snow was a favorite to sit on the Iron Throne at the end of Game of Thrones due to his secret Targaryen heritage, his true bloodline actually foretold a different outcome. In the Game of Thrones series finale, the Iron Throne was melted and in its place a new government overseen by King Bran the Broken.
Jon's was introduced as the bastard of Ned Stark and an unnamed woman, but his parentage was slowly revealed to be so much more important than that. Over seasons, the truth of Jon Snow's birth - he was the son of Rhaegar Targaryen and Lyanna Stark, Ned's sister - was revealed, through both Bran's visions and Sam's discoveries at the Citadel (aided, of course, by Gilly). It seemed as though this would become a huge plot point due to it making him first in line to rule, but while it certainly upset Daenerys, in the end, Jon was sent back to the Night's Watch, bringing his story full circle.
Some fans may have been upset that Jon ended up North of the Wall with the Wildlings, rather than ruler of the Seven Kingdoms, but those who know their Targaryen history (like this Redditor) were always aware that dark-haired Targaryens just don't end up on the throne. The majority of Targaryens have particularly distinctive coloring - white blonde (almost silver) hair, and purple eyes. However, Jon Snow is dark-haired, with dark grey eyes (Kit Harington's eyes are dark brown). In-universe, this is clearly the Stark genes coming through, but Jon is not the first Targaryen to lack the silver/purple coloring - and not the first of those who was in line to the throne and missed out.
Rhaenys Targaryen (daughter of Aemon) had black hair and was in line to the throne before the Dance of the Dragons began, but a ruling council disregarded her claim due to her gender. During the Dance of the Dragons, Jacaerys and his brothers Lucerys and Joffrey were the sons of Rhaenyra Targaryen - one of the major claimants during the dance. All three were brown haired, and all three died during the war. Baelor, son of Daeron II, was not only heir but Hand of the King, but was killed in a tourney before ever taking the throne. His own son, Valarr, would have been heir after him, and also had brown hair, but died of the Spring Sickness. Two more dark-haired Targaryens also came in line to the throne, but lost it: Daeron the Drunken (who died of a pox) and Duncan the Small (disinherited for marrying a commoner).
Hair color means a lot in Game of Thrones - the Lannister children born of incest (Joffrey, Myrcella, and Tommen) were all golden-haired, like most Lannisters. Most Starks are dark-haired, Gendry and the Baratheon bastards were dark-haired, and the Targaryens are mostly silver-haired. Therefore, it's easy to see how Jon Snow's hair color didn't bode well for him; as a rule, Targaryen monarchs are silver-haired, and those with darker hair - even heirs with dark hair - fail to become ruler.
Of course, George R.R. Martin is using fantasy genetics, rather than anything based in fact. There are exceptions to all of these hair color rules; Sansa with her red hair, for example. While hair color can be meaningful, it doesn't have to be. Jon Snow's hair may have been a sign from the start that the Iron Throne would always be out of his reach, or his ending may have had a simpler explanation: that a return to the Wall where he started out is circular storytelling, and that it was more bittersweet (and less obvious) than having him take the throne in the end (conversely, there is previous history back Bran the Broken). After all, Daenerys looked as Valyrian as could be and failed to take the throne. As much as Game of Thrones fans may have wanted to see Jon rule, he made it more than clear in the final season that he simply didn't want it.