The question of Jon Snow's legitimacy in Game of Thrones is a valid query and There are some important reasons why Ned Stark never went through this. Much of Jon Snow's arc in Game of Thrones It comes from the way he was raised. Despite being raised in a noble castle, Jon was an outcast for being the bastard of the Stark family; a stain on the immaculate nobility of Lord Eddard Stark. Still, both Game of Thrones and the A Song of Ice and Fire the books portray Ned and Jon as having a decent relationship, so it's curious why it was never legitimized.
The legitimization process would have resulted in Jon Snow being renamed Jon Starkmatching him with similar possibilities to Bran. In the House Stark family tree, Robb is still older than Jon, so he would have inherited Winterfell. However, as shown by second sons such as Daemon Targaryen and Stannis Baratheon, animosity often arises towards the firstborn's birthright. After all, all it takes is one bad slip down the stairs for the firstborn to die and the secondborn to inherit everything that was theirs, which means that the difference between Jon being Stark and Snow has an important meaning.
Legitimizing bastards is very uncommon in Westeros – and requires a monarch to do it
Ned Stark wouldn't be able to legitimize Jon without Robert Baratheon's permission
The first issue to be discussed regarding the process of legitimizing a bastard is the logistics of doing so. Ned Stark has immense power as Lord of Winterfell and Warden of the North, but legitimizing a bastard is something only the monarch of the Seven Kingdoms can do. This is why Stannis Baratheon, a self-proclaimed king, offered Jon legitimization in Game of Thrones season 5. Although Tommen and Cersei disputed the legitimacy of his legitimization, it would have been true if Stannis had somehow won the War of the Five Kings.
Before Ned Stark's death and the events of Game of Thrones, it would be up to Robert Baratheon to legitimize Jon. Given the close bond between Ned and Robert, he might be willing to do so. There is really no process necessary for a bastard to be legitimized; it is only up to the king to decide. Roose Bolton received permission from the crown to legitimize Ramsay Snow in Game of Thrones Season 4, and this was essentially a gift for Roose's involvement in the Red Wedding.
Ned asking Robert to legitimize Jon would have been a big risk
Ned doesn't want any attention to Jon's parentage, especially Robert's.
Knowing what Game of Thrones The season 6 finale reveals Jon Snow's ancestry, any connections between Robert Baratheon and Jon are a huge risk to his safety. Ned's main goal in raising Jon is to protect his nephew, after making a promise to his beloved sister, so he doesn't want to create any unnecessary risk of Robert finding out who his mother was. Knowing how much Robert detests Rhaegar Targaryen, he would no doubt demand to kill Jon if he realized who he really was.
Asking Robert to legitimize Jon would likely start that conversation, and Ned could easily slip up and get caught lying.
Robert asks about Jon's mother in Game of Thrones season 1, episode 2, although he believes the mother was Wylla, a prostitute they met in the south during the war. Ned is able to deflect and divert the conversation from the topic because he doesn't want to continually lie about a woman who possibly never existed. Asking Robert to legitimize Jon would likely start that conversation, and Ned could easily slip up and get caught lying.
Drawing Robert's look at Jon Snow is exactly what Lyanna wanted to avoid. Happily, no one really pays attention to the bastards of Westerosi high societyso Jon is essentially able to hide in plain sight, out of Robert's attention. This may mean bullying and ostracism for the boy, but at least he's still alive and growing up in a place where Ned can care for him. Jon joining the Night's Watch is just another step away from Robert's potential wrath.
Ned legitimizing Jon Snow could have caused problems for House Stark
“Jon Stark” would create potential succession conflicts
In addition to putting Jon's safety at risk, legitimizing him creates potential problems within House Stark. First, Catelyn despises Jon as a reminder of Ned's infidelitybelieving he was a stain on her honor. Although she regrets her hatred of him, seeing Jon legitimized wouldn't help. Having Jon potentially compete with Robb for claims to Winterfell and other lands and titles would be a huge insult to her. It's hard to say whether he would have more rights than Bran, but it would certainly create a controversy that she would prefer to avoid completely.
Westerosi society has rules for succession, but they can be questioned. For example, Ramsay Bolton is legitimized in season 4, but when Roose Bolton and Walda Frey announce their new child as a boy, it poses an instant threat to him. Despite being older and legitimized, Ramsay was still born a bastard. Regardless of what is considered right under the law, these situations create conflict. Jon is not power-hungry by nature, but could have become so if he was legitimized and considered himself worthy of inheriting more.
“Jon Stark” wouldn't fit Ned Stark's honor
Ned Stark would not want to perpetuate a lie
Ned Stark considers few things above his own honor, nearly endangering his daughters' lives in season one before finally agreeing to proclaim Joffrey king. With that in mind, having a bastard is a stain on your honor, but it's not entirely uncommon. So many lords of Westeros have bastards that it hardly reflects badly on the father. Ned takes things a step further, however, as he's lying about Jon being a bastard. Technically, Jon isn't even a Stark. He would be considered a Targaryen under Westerosi laws.
For Ned, having Jon legitimized would further contribute to this facade, something Ned probably wouldn't want to do. He is not a man who likes to lie or weave webs of misconceptions. Pretending that Jon is somehow a legitimate son, at least under the law, would probably be a step too far for what he's willing to do. Her actions with Jon before Game of Thrones will keep a promise to his sister, but he won't do anything more than is necessary if it means being deceitful.