If Jon Snow is indeed the son of Rhaegar and Lyanna, then - unlike many of the other bastards around (including Robert's progeny), he's noble on *both* sides. And of course he's quite literally serving the realm as a whole rather than taking any side in its internal disputes.
Of course if he's Eddard's own son as per Eddard's official statements, then his mother is not disclosed, unless it's either (a) Lady Ashara Dayne, which he denies absolutely, or (
the lowborn Wylla, whose name Eddard mentioned to Robert early on but has since been revealed only as Jon Snow's wetnurse (though of course this does not preclude her also being the mother.)
Daenerys is technically "legitimate" as a daughter of Aerys, but she's a child of incest (and probably incestuous rape), and much of the story revolves around how the Targaryens had no real respect for either god or man, and how incest is cursed by gods and men. In any case, the Targaryen claim to the throne was no more legitimate than Robert's, both being based on conquest, force and fear, except it was three hundred years older. (And the Stark claim to be King in the North is older still.) And that has been three hundred years in which they have had a rare few good kings, a whole lot more bad kings, a whole load of incest which would have been prohibited with any other family, and several more buckets of madness which would have probably had the Targs dethroned long ago if it wasn't for the fear of what they might be able to do (fear of the dragons would persist long after the last dragon died, and it takes the defiance of Robert's Rebellion - and possibly the preceding Defiance of Duskendale - to break that fear.)
Aegon, if he is who he is purported to be, has a claim both via descent from Rhaegar and Aerys, would come ahead of Dany in the line of succession, *and* doesn't have the stigma of incestuous parentage since his mother was Elia Martell. However, there are questions about his true parentage.
Stannis is the legitimate heir of Robert by any definition *other* than the definition of conquest. However, even if he were accepted as King by the whole kingdom, it would only postpone the dispute rather than resolve it - his daughter being disfigured by greyscale, may be still sick, will never find a husband, and he seems to be sufficiently estranged from his wife (and, indeed, from the whole business of being married - he really ought to be one of life's permanent singletons) that he will not sire a son by her, yet he does not formally divorce her to remarry either: so Stannis will not have a legitimate son, and thus would never have had any heirs but his brother. And Renly is now dead - not that *he* was likely to sire any heirs either, given his own preferences (which are not just a creation of the TV show, since Stannis himself alludes to them... Renly: "She came to me a maid, you know". Stannis: "And in your bed she's likely to remain that way." Evidently Stannis knows well enough that Renly does not prefer women.) Although he might have eventually managed to have one child by Margaery, but as things turned out, he didn't.
So, even were there a peaceful resolution to the fight for the Iron Throne, there would be no heirs beyond Stannis and Renly for the Baratheon family, so who would be next in line? Evidently a lot of arguments, and possibly another fight.
To be honest, it would have been better if the Lannisters had taken the throne "by right of conquest" as Robert did, and put Tywin on the throne himself, rather than try to carry on with the pretence that Joffrey was a Baratheon and rule through him. Mind you, the reason Tywin didn't do that was almost certainly that otherwise, that would make Tyrion his heir (which actually might not have been a bad idea for the kingdom), since Jaime had joined the Kingsguard and thus disqualified himself from noble position. Alternatively, Tywin could have taken the throne, officially disowned Tyrion (as Randall Tarly did with Samwell even though Samwell was the elder son) while passing him off with the job of Hand or Master of Whispers, and named Kevan his heir, and after him, Lancel. Disowning Tyrion and giving him a job would actually have been kinder than keeping him around and continually humiliating him.
Then Eddard would have had to be thrown down (imprisoned and possibly sent to the Wall - Tywin would have chosen the latter, to prevent a Stark/Tully rebellion), but the Kingdom would at least have an able ruler (Tywin), an able Hand (Tyrion), and a clear line of succession (Tywin I, Kevin I, Lancel I). Gregor Clegane could even then have been safely sacrificed to the wrath of the Tullys and Martells for his misdeeds, the Tyrells bought off with a marriage to Lancel who would then be crown prince (in turn, depriving Renly of his biggest supporter), and the Starks could be left alone in the north (with Eddard at the Wall and Robb in Winterfell).
You could actually say that the biggest fool in the whole series is Tywin - for exactly that decision, choosing to try and rule through the cloak of legitimacy rather than claiming right of conquest and starting a new dynasty which could theoretically end up with a "legitimacy" of its own if it lasted long enough. When you're going to try and steal power, sometimes it's better to admit it rather than pretend there's any right on your side.
Edited by JLE, 24 March 2013 - 03:21 PM.