The prototypical Mary Sue is an original female character in a fanfic who obviously serves as an idealized version of the author mainly for the purpose of Wish Fulfillment. She's exotically beautiful, often having an unusual hair or eye color, and has a similarly cool and exotic name.
This would be the Targs but also Tully looks are seen as exotic by the fandom.
She's exceptionally talented in an implausibly wide variety of areas, and may possess skills that are rare or nonexistent in the canon setting. She also lacks any realistic, or at least story-relevant, character flaws — either that or her "flaws" are obviously meant to be endearing.
This can be Sansa. He tries to make Jon non threatening so he's average. He's never the best at anything. That can be typical of certain heroes though that are meant to be relatable in order to make them more sympathetic.
She has an unusual and dramatic Back Story. The canon protagonists are all overwhelmed with admiration for her beauty, wit, courage and other virtues, and are quick to adopt her as one of their True Companions, even characters who are usually antisocial and untrusting;
Jon has a dramatic back story. The overwhelming admiration is for Sansa and Dany but the latter is getting mad/Aerys PR already lately.
Jon did gain allies in situation where it's implausible and there are females who want him.
if any character doesn't love her, that character gets an extremely unsympathetic portrayal. She has some sort of especially close relationship to the author's favorite canon character — their love interest, illegitimate child, never-before-mentioned sister, etc.
This definitely applies to Jon. Catelyn and Tyrion.
Other than that, the canon characters are quickly reduced to awestruck cheerleaders, watching from the sidelines as Mary Sue outstrips them in their areas of expertise and solves problems that have stymied them for the entire series. (See Common Mary Sue Traits for more detail on any of these cliches.)
The fandom certainly believes this will happen to the point where many act like other characters' roles are in the story solely to how they can either be or not be of help to the oh so heroic and kingly Jon Snow.
Rhaegar is probably a worse offender. Especially if fans get their wish and Elia has no problem so that fans can use that to absolve him.