Azor Ahai/The Prince That Was Promised/The Last Hero will not be anybody in the series. Not Jon, not Stannis, not Aegon, not anybody. They will not appear.
I'm basing this on two of the most familiar tales/religious beliefs/myths known to people from Western Europe, and Britain/France in particular: those of King Arthur and of course, Jesus. You don't need to be Christian to know the prophecy about how Jesus will return in the world's greatest moment of need, that he rose from the dead on Easter after being crucified, etc. Likewise, King Arthur is also prophesied to return in Britain's greatest moment of need. King Arthur is probably the closer comparison for the return of some awesome guy in ASOIAF due to the parallels of the uber sword, and the general issue that Jesus apparently wouldn't hurt a fly, but they both serve the notion of different peoples holding beliefs that if shit hits the fan in the future, a past hero will return to bring salvation (I say different peoples because King Arthur predates Christianity in Britain, and is a Celtic hero, despite his Christianization in later adaptations of the myth). One of the titles afforded to Arthur is "The Once and Future King," which is remarkably reminiscent of "The Prince That Was Promised".
There have been countless supposed "returns" of these two men throughout history, with Jesus pretenders more common than Arthur ones. But in the time period of British history which GRRM draws on fairly heavily for his series (the Wars of the Roses), the return of King Arthur is a big question that is posed, especially at the very end, when Henry VII takes the throne in 1485. Arthurian legend was dramatically stimulated by the publication of Thomas Malory's Le Morte d'Arthur in 1485, being the most complete rendition of the mythology since Geoffey of Monmouth's History of the Kings of Britain in the 12th century. Malory's version is the one we grow up with, as its enormous popularity caused it to be the main source of almost every Arthurian tale published in the years after. Malory wrote the story while imprisoned, as he watched England being torn apart during the Wars of the Roses, and it is imbued with many thematic and tonal elements that reflect those events.
Now, when Henry VII comes to the throne, he has his court historians compile his ancestry, and they trace his heritage back to King Arthur, and place the original location of the city of Camelot at Winchester, the location supported by Malory's text. When it is clear that Henry's wife, Elizabeth of York (York as in the White Roses of the Wars of the Roses; Henry is descended from the Red Roses, i.e., the Lancasters... As York is in the north, and the Lancaster territories are comparatively south, I suppose a case could be made for Henry VII and Elizabeth of York being Rhaegar and Lyanna) is pregnant, Henry proclaims that she will birth a son who will bring a golden age to Britain, and he moves his entire court to Winchester (a.k.a. "Camelot") so that his son can be born there in honour of his ancestor, King Arthur, whose great feats the new heir apparent would equal. Sure enough, a baby boy is born in 1486, and Henry duly dubs him Arthur Tudor. Everyone rejoices, because Arthur has returned, and when he comes to the throne, he will heal the hurts of Britain, and be a good and gracious king, because you know, second coming and all.
Arthur grows up, and in one of the most promising of political alliances, marries Catherine of Aragon, a Princess of Spain, at the age of 15. About five months later, he dies of some sickness, which is not known, but possibly tuberculosis, or the sweating sickness. As a result, his younger brother, Henry, is made the new Prince of Wales (heir to the throne), and eventually Henry marries Catherine, his brother's widow. When Henry VII dies in 1509, the prince who was not promised is crowned King Henry VIII (as in the one with all the wives).
Now, we can try to fit all of the characters into the actual history, in which case it seems that Aegon is Prince Arthur Tudor, and then his younger brother would be Jon, who currently is repping Henry VIII (assuming that both are real); Rhaegar believed that his first son, Aegon, was TPTWP, just as Henry VII believed Arthur was the Once and Future King. In both situations, that beacon of hope is snuffed out before they reach the throne. Aegon has possibly escaped his infantile fate, but there is also the strong argument that he's fake, and I'm not going to debate for either. But if we are following the historical model I've set out here, Aegon's probably going to marry Daenerys, die of Westeros' version of sweating sickness or TB (grey scale or the Pale Mare) shortly thereafter, and then Jon marries Dany, his brother's wife. Re: IF we follow the historical model.
Regardless of whether Aegon and Jon are Arthur and Henry, or whether Aegon or Jon is supposedly TPTWP, or AA, or whatever, the history that I'm pushing here is that there never is a second coming of King Arthur. People prophecy that he's going to come back, they name princes after him, saying that they are Arthur reborn, and it is a major feature of the Wars of the Roses and the victorious Tudor dynasty. But Arthur never comes back, and I don't think AA/TPTWP will either. It doesn't mean that Aegon or Jon or whoever won't come to the throne, but the one that does won't be the returned hero.
OH and King Arthur's coat of arms? A red dragon. Just saying...