As I've pointed out [in a similar post](http://movies.stackexchange.com/a/33565/11244) on Movies&TV:

<h1>We do not know; it is not explained... yet.</h1> 

**She wants him bad**  
In the books [**Melisandre**][1] *is* very interested in [**Jon Snow**][2], believing that there is *some* power in him:

<sup>[from *A Dance With Dragons*]</sup>
>!“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.”<br />I am not a wolf, he thought. “And how would I do that?”<br /><sup>*-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 29 (Jon VI).*</sup>

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 then goes on to say the whole "power to make shadows" line:

>“I can show you.” Melisandre draped one slender arm over Ghost, and the direwolf licked her face. “The Lord of Light in his wisdom made us male and female, two parts of a greater whole. In our joining there is power. Power to make life. Power to make light. Power to cast shadows.”<br /><sup>*-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 29 (Jon VI).*</sup>


She's also interested in [**Mance Rayder**][3] and his [son with Dalla][4], 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**][5] (or [**Edric Storm**][6] 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**][7] were [Kings In The North][8], it wouldn't have mattered *per se* that Jon is [**Robb**][9]'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][10]. 
 - 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**

 - *She* doesn't even know. There are hints at Jon Snow actually being [the one][11] she is looking for instead of her assumed **[Stannis][12]**:

 > 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.<br />
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.<br /><sup>*-A Song of Ice and Fire: Book 5 - A Dance With Dragons, Chapter 31 (Melisandre I).*</sup>

 - Couple this with the fact that [Melisandre is a little bit of a rogue](https://scifi.stackexchange.com/a/67163/21267); "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


  [1]: http://awoiaf.westeros.org/index.php/Melisandre
  [2]: http://awoiaf.westeros.org/index.php/Jon_Snow
  [3]: http://awoiaf.westeros.org/index.php/Mance_Rayder
  [4]: http://awoiaf.westeros.org/index.php/Aemon_Steelsong
  [5]: http://gameofthrones.wikia.com/wiki/Gendry
  [6]: http://awoiaf.westeros.org/index.php/Edric_Storm
  [7]: http://awoiaf.westeros.org/index.php/House_Stark
  [8]: http://awoiaf.westeros.org/index.php/King_In_The_North
  [9]: http://awoiaf.westeros.org/index.php/Robb_Stark
  [10]: https://scifi.stackexchange.com/q/32314/21267
  [11]: http://awoiaf.westeros.org/index.php/Azor_Ahai
  [12]: http://awoiaf.westeros.org/index.php/Stannis_Baratheon