The Targaryen blood in House Baratheon, both in their founding and in their marriage to a Targaryen princess a couple of years back, is what gave them their claim to the throne. With House Baratheon extinguished (RIP) in the show, Daenerys really is the only candidate left with any real legitimacy. And there's some precedent about no women on the throne either, so if you reject her (though I think she well have more support in Westeros than many suspect) you have to go back and find some legitimate Targaryen descendants.
As far as I can tell from the family tree in TWOIAF, and assuming the claim can pass through the female line, this give us...the Martells, throught the marriage of Daenerys Targaryen (not the current one) and Maron Martell. House Martell is now apparently extinguished in the show as well (RIP) because let's avenge Oberyn horrific but technically legal death in a trial-by-combat by murdering his family (?!???) who are also our family, making us kinslayers. But I digress.
At this point in the family tree, all the incest starts confusing the fuck out of me, and I just give up and conclude that Brown Ben Plumm through Elaena Targaryen's line is the true heir. But he's not in the show, and honestly all the other noble titles in Westeros let a man's daughter inherit before his brother anyway, so let's just give the throne to Daenerys and have her go mad and melt it down into Valyrian steel to fight the White Walkers or something. Like honestly she may be power hungry but I don't she is any worse than say Aegon the Conqueror. Everyone loves Aegon the Conqueror and he didn't even have any legitimacy or precedent for what she did. Dany is just trying to re-establish a dynasty that, like it or not, ruled a continent for 300 years. No small task, nor one that can be completed without bloodshed.
Anyway, during all this I noticed off on the left of the Stark family tree there are family lines that trail off into "issue" but would still have the Stark name. I wonder if these cousing Starks are off somewhere during the series, but surely they would be mentioned?