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• She doesn't even know. There are hints at Jon Snow actually being the one she is looking for instead of her assumed Stannis:

There was no one, even in her order, who had her skill at seeing the secrets half-revealed and half-concealed within the sacred flames.
Yet now she could not even seem to find her king. I pray for a glimpse of Azor Ahai, and R’hllor shows me only Snow.
^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 31 (Melisandre I).}

• Couple this with the fact that Melisandre is a little bit of a rogue; "she is on her own mission and has her own agenda"
• Something is telling her that she is in the right place and on the right track

• She doesn't even know. There are hints at Jon Snow actually being the one she is looking for instead of her assumed Stannis:

  There was no one, even in her order, who had her skill at seeing the secrets half-revealed and half-concealed within the sacred flames.
  Yet now she could not even seem to find her king. I pray for a glimpse of Azor Ahai, and R’hllor shows me only Snow.
  ^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 31 (Melisandre I).}

• Couple this with the fact that Melisandre is a little bit of a rogue; "she is on her own mission and has her own agenda"

• Something is telling her that she is in the right place and on the right track

As I've pointed out in a similar post on Movies&TV:

As I've pointed out in a similar post on Movies&TV:

• She wants his blood, because of R+L = J.
• She wants his blood, because he is a Stark, and the Starks were Kings in the North.

• Couple this with the fact that Melisandre is a little bit of a rogue; "she is on her own mission and has her own agenda"
• Something is telling her that she is in the right place and on the right track

• She wants his blood, because of R+L = J.
• She wants his blood, because he is a Stark, and the Starks were Kings in the North.

• Couple this with the fact that Melisandre is a little bit of a rogue; "she is on her own mission and has her own agenda"
• Something is telling her that she is in the right place and on the right track