Nothing in 'Game of Thrones' ever happens without a reason, even when it comes to naming children. Recent speculations on Reddit have come up with the theory that both the Stark sisters, Arya and Sansa, were named after their great-grandparents - and probably after some of the founding members of the Stark house.
A recent discussion on Reddit spun over the naming of the Stark children and it turns out that each Stark kid (including Jon Snow) was named after people who were extremely significant in the history of House Stark. Commenting on the introductory statement, "I was just thinking about it and realized that all four Stark boys are named after people who were important to Ned (Robert Baratheon, Jon Arryn, Brandon Stark, Rickard Stark) but at least in the show we don't know of any important characters who share the girls' names or variations of them", one Reddit user claimed that both Arya and Sansa were named after Ned's and Cat's grandparents, respectively.
The user, named Chaost, commented, "Ned's grandmother was named Arya Flint. There's a Sansa higher up, but it's harder to see the relationship...So Cat's grandmother was also named Sansa Leeford, so it looks like naming after grandmothers is the connection for both. It makes sense if it was Cat's idea to do do it first after her own grandmother to follow suit with Ned's. Well, case closed."
While this theory is quite substantial in itself, it could also be said that the present Lady of Winterfell was named after one of her predecessors who was also a Lady of Winterfell.
As the second daughter of Rickon Stark, heir to Lord Cregan Stark, and Jeyne Manderly, Sansa Stark was married to Lord Jonnel Stark, also known as One-Eye, who served as the head of the House Stark in the latter half of the second century after Aegon's conquest. On the other hand, another theory states that the youngest daughter in the Stark family, who went out on a quest and returned as 'a girl has no name', was named after Jon Arryn. As a respectable nobleman, Jon served as the Warden of the East and Defender of the Vale and was the Lord of the Eyrie, Lord Paramount of the Vale, Warden of the East, and the head of House Arryn.
His death, which happened under some mysterious circumstances, took place right before the current events that the show is based on. However, even more than Arya being the rightful heir to Jon's name, it wouldn't be wrong to assume that our favorite bastard, Jon Snow, too was named after him. Almost carrying out the legacy of the Warden of the East, Snow himself has served as the Warden of the North, Lord Paramount of the North, Lord of Winterfell. The last season saw him being hailed as the King in the North, and probably very soon he will be heading for the Iron Throne, provided the Mother of Dragons already doesn't claim it.
Although all of it is a mere assumption and there cannot be a full assurance about the names, probably Bran is the only one who has a legitimate origin to his name. The three-eyed raven has been the subject of several controversies which consider him to be the actual Night King specifically after Bran was touched by the supreme leader of the White Walkers which left a mark on his arm. Bran was named after Brandon Stark, the heir of Rickard Stark, the Lord of Winterfell, and the older brother of Eddard, Lyanna and Benjen Stark. He was killed by strangulation by the Mad King when he protested the kidnapping of Lyanna by Prince Rhaegar Targaryen.
In a conversation between Ned Stark and Petyr "Littlefinger" Baelish, it was revealed that Brandon had once nearly disemboweled Littlefinger in a duel, in order to defend Catlyn's honor. Remember Ned's remark when Littlefinger recalled, "All too well. I still carry a token of his esteem from navel to collarbone." Ned had simply snapped, "Perhaps you chose the wrong man to duel with." (Bang On!)