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

We do not know; it is not explained.

She wants him bad
In the books Melisandre is very interested in Jon Snow, believing that there is some power in him:

^{[from A Dance With Dragons]}

“You think so?” She knelt and scratched Ghost behind his ear. “Your Wall is a queer place, but there is power here, if you will use it. Power in you, and in this beast. You resist it, and that is your mistake. Embrace it. Use it.”
I am not a wolf, he thought. “And how would I do that?”
^{-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 29 (Jon VI).}

She does not explain exactly where she believes his source of 'power' to be from, but she definitely feels it. Jon though, immediately assumes that it's because she thinks he's a "Wolf" (Starks are often referred to as "Wolves").

She's also interested in Mance Rayder and his son with Dalla, for the same "King's Blood" reasons; but that's a different point.

If you believe that King's Blood is what carries power, then theoretically; Yes, Melisandre wants Jon Snow's blood.

Bastard or not
This is why Melisandre was interested in Gendry (or Edric Storm in the books). Even though Gendry is a 'bastard', she still believes that the Blood (and power which comes with it) of the King runs in him.

Stark contrast
You see, historically, the Starks were Kings In The North, it wouldn't have mattered per se that Jon is Robb's brother, it matters that Jon is of Stark descent; so Yes Jon has the blood of Kings running through him as he is A Stark.

Whether R+L=J, or not; since L is in fact a Stark.

So let's break-down the possible reasons Melisandre could want Jon:

1. King's Blood

• 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

2. Red Blooded Woman

• She is feeling, lonely and needs some attention

3. Unknown

• Unlikely, since there doesn't seem to be any more reason for anything else