The Game of Thrones wikia simply refers to it as Shadow. In an interview with Carice Van Houten (Melisandre) and Liam Cunningham (Davos) Carice refers to it as a "Shadow Baby".
Can you single out your weirdest or most challening scene?
Carice: Giving birth to the shadow baby would be the answer for both of us. We were there together.
Liam: Yeah, I had a few pains… I felt your pain, darlin’.
SEASON 4 INTERVIEW: CARICE VAN HOUTEN & LIAM CUNNINGHAM
If your character meets an ending like Ned Stark… what memory would you best remember from your time on Game of Thrones?
Carice: The shadow baby! But mostly just being around Stephen [Dillane] and Liam. They couldn’t be more different as actors, and as people. You could see a sitcom about these two guys. It’s a very funny threesome, I would say.
Bryan Cogman one of the writers of the show has also referred to it as a "Shadow Monster".
When was it decided that they were finally going to appear? Well, you had a stand-in for Selyse in the opening of S2, and I think Melisandre makes a passing reference to her, but this is quite a bump up in visibility.
“It was an evolving thing. Season 2, as ever, we had to be judicious about how many characters to introduce. It was decided that, dramatically, it made sense for the S2 Stannis scenes to involve this triangle of characters, if you will—Davos on one side, Melisandre on the other, Stannis being pulled between the two. In the book, you have Selyse on the Mel side but we decided to pull her out of the political discussion, at least for now. But when the decision was made to dramatize the conception of the shadow monster…”
SEASON 3 INTERVIEW: BRYAN COGMAN
Davos seems to refer to the monster as "Shadow" or "a horror" in his POV chapters from the books too.
He could not behold the fiery heart without thinking of the shadow Melisandre had birthed in the gloom beneath Storm's End.
A Clash of Kings, Davos III
"The Mother. She blessed me with seven sons, and yet I let them burn her. She spoke to me. We called the fire, she said. We called the shadows too. I rowed Melisandre into the bowels of Storm's End and watched her birth a horror." He saw it still in his nightmares, the gaunt black hands pushing against her thighs as it wriggled free of her swollen womb. "She killed Cressen and Lord Renly and a brave man named Cortnay Penrose, and she killed my sons as well. Now it is time someone killed her."
A Storm of Swords, Davos II