One thing that most fans who have discussed this issue seem united on is that Stannis is not the PtwP, despite Melisandre’s claims or beliefs. Maester Aemon, like Salladhor Saan, cast doubts on the authenticity of Stannis’s Lightbringer (II: 115. III: 886). While the rest of her prophecy (II: 110. III: 289) bears a superficial resemblance to Stannis’s own circumstances, other candidates have greater support among the fan community.
Jon or Aegon (if alive) are seen as likely possibilities due to Rhaegar’s statement that a son of his was the promised prince (see Prophecies). However, the details of the prophecy do not seem to support this. While Jon’s confrontation with the wight in the Old Bear’s chambers (I: 473-474) did involve fire around the time that the comet was first noted, there’s no salt. Aegon, being a complete cypher, is even more unclear although one can assume that if it’s Aegon, he certainly cannot be Samwell or Edric Dayne, whom the prophecy does not match whatsoever.
The next most popular theory is that it’s Daenerys, which has gained strong support following Maester Aemon’s revelations (IV: 520). The smoke and salt may be a reference to the funeral pyre and the salt of her tears, or it may refer to the fact that she was born on Dragonstone. Daenerys also seems to notices the comet for the first time as she lights the pyre. Rhaegar’s belief that one of his children was the prince who was promised may thus be in error on his part. Following this theory, it’s believed that Lightbringer for this reborn Azor Ahai may in fact be her dragons. One argument against this was that she is a princess, not a prince, but "prince" can have a gender-neutral usage ("the princes of Europe") even in English, and in any case Maester Aemon makes it plain that the original Valayrian word may have had indeterminate gender because of the fact that dragons change between male and female.
The last of the reasonable candidates is Ser Davos Seaworth, whose plunge into the fiery Blackwater (II: 610) while the comet still hung in the sky could possibly fulfill the prophecy. This is, however, the only thing about him that fits, and so it’s not a popular theory.