A very difficult one to answer. I'd say officially, we don't know:
Boring Occam's Razor stuff
- We don't know if there is a R'hllor1
- We don't know if this R'hllor has or needs a Champion
- Melisandre thinks that she is R'hllor's Champion, but she is somewhat misguided and rogue2.
- We don't know if there is in fact a Great Other3
- We don't know that this Great Other has or needs a Champion4
1. There is no clear evidence that the gods in the A Song of Ice and Fire universe are real. It is merely a belief of the inhabitants.
2. We have only a single view-point chapter of hers - A Dance With Dragons: Chapter Thirty-One (Melisandre I); and there is no clear evidence of her speaking to or being told by Rh'llor that she is in fact his Champion.
She's also gone somewhat rogue and refers to Stannis as her Champion.
George R. R. Martin points out in a recent interview that she is essentially operating on her own volition with Stannis:
[Interviewer] Why did Melisandre seek out Stannis? Did she see him in her flames and decided to seek him out on her own, or is she on a mission on behalf of the red priests? It doesn't seem at any point as if the latter is the case, when you compare to Moqorro who has been sent out by the priesthood.
[GRRM] You're right. Melisandre has gone to Stannis entirely on her own, and has her own agenda.
And from her viewpoint chapter:
The wooden man she had glimpsed, though, and the boy with the
wolf’s face … they were his servants, surely … his champions, as Stannis was hers. [emphasis mine]
-A Song of Ice and Fire: Book Five - A Dance With Dragons, Chapter Thirty-One (Melisandre I).
She's a funny kid.
3. In the same way as number 1, there is no evidence of there being a Great Other (god or otherwise). This seems to be a belief of the followers of The Lord of Light.
4. Melisandre is the only person to refer to The Great Other and his need for a Champion. Hardly reliable and concrete evidence.
What says ol' George?
We must be careful with what we listen to in a character's point of view. Much is written to incite intrigue and mystery. Purposefully, George R. R. Martin has left these "magical" aspects of his story vague and unexplained. It is to be left up to your own imagination and decisiveness. If you were to ask him, I'd bet his answer would be "well what do you think?" or "wait till the next book".
All we can do is speculate...
Woohoo! Fun speculation time
For this part to work, we need to make the following assumptions:
- There is a Rh'llor
- Conversely, there is a Great Other
- Both have/need a Champion
- Melisandre is Rh'llor's Champion
The hot favourite for this position. He's lost. Currently delved deep in the North, the heart of Winter, The Night's King and everything E-ville. Has shown signs of using people to his advantage (mind melding with poor old Hodor). He's currently being mentored by Brynden "Bloodraven" Rivers (The Three Eyed Crow) - another person known to use nefarious means to an end.
This is my favourite. The Three Eyed Crow. The Last Greenseer. Bloodraven. Lord Commander of the Night's Watch (defected). Hand of The King (defected). The riddle goes:
How many eyes does Lord Bloodraven have? A thousand eyes, and one.. Bloodraven is unfortunately no longer himself. He's turned into some weird cave-dwelling-tree-dude. He's been suspected of commissioning Jojen Paste.
I believe that it's neither of them. If there are Gods, then I believe that there are the "Old Gods". Bloodraven's "abilities" seem to be very much aligned with the powers of the Old Gods. The Old Gods seem to be much removed from The Great Other (probably adversarial on some way or other, but still different). Hence, I believe that if anything, Bloodraven and Bran are agents of The Old Gods.
Let's say Jon Snow dies. Melisandre brings him back, but somehow distorts him and turns him evil (it's very possible — GRRM has said that the dead don't come back "new and improved", they come back as "changed" and become "vengeful spirits"); he could easily end up being The Great Other's Champion. See my discussion thread on this theory on Westeros.org.
Yes, Melisandre herself. She might not realise it, but she herself is doing some messed-up stuff. Something we consider "evil". She may very well (inadvertently) be The Great Other's Champion.
Stannis The Mannis
There are many, many characters, some even seemingly un-important until later on, it could very well end up being Bronn, or Euron (I'd doubt it), or Stannis or Daenerys; but unfortunately we will not know until it happens.
All in all, there's just not enough information to conclude effectively at this point; so hang tight, we will soon[ish] find out!