We no doubt have several rounds of murder, battle, doinking and intrigue to go but who, from a purely political point of view, has the most legitimate right to claim sovereignty over Westeros?
Cersei is the de facto sovereign at the moment. Her claim may not be the strongest in law (de jure) but she has consolidated and maintained power since killing or otherwise dealing with her immediate challengers. Her will has prevailed and is obeyed. For now. However, she is not quite absolute in her control. She cannot claim, Louis XIV-style, that she is the realm, for the realm remains too divided. Could she achieve total power? That appears to be the aim but even absolute monarchies have to be bounded by a social contract: individuals will give up some freedoms if their remaining rights are protected. Perhaps ex-Maester Qyburn will be Cersei's Thomas Hobbes, setting out a vision for imposing rules that leave the populus secure and just free enough.
Danaerys and Tyrion strike me as John Locke fans, though, seeking more legitimate forms of rule based on consent. Throughout her proto-nation-building across the plains and cities of Esteros, the Mother of Dragons has certainly behaved dictatorially but with a constant awareness of the need to promote the public good. Her version of the social contract has been less imposed from above and more won from below (for example, by the freeing of slaves). She will need a little more to establish her legitimacy amongst the more sophisticated populations of Westeros, though. So expect to see the diminutive Hand of the Queen spinning her de jure claim to the throne as the heir to the usurped Targaryen dynasty, as well.
|Grand Maesters Hobbes and Locke|
Can this popularity be spread to the other kingdoms? Well, leading a victory in a great war against an enemy posing an existential threat would do just that. Just so happens that Lord Snow has a bit of previous there.