It would be kind of an anti-climax if the infant just became a wight, after all that build-up. Any dead thing becomes a wight, so this long ride, the ice pedestal and ceremony would be overkill if all they meant to do was create a wight. So this is clearly something more.
I think this is the TV-show fleshing out the story from the books where it is merely implied that Craster and his sons have something to do with the Others (the White Walkers). I think the most clear hint in the books is the scene where Sam and Gilly are fleeing from Craster's, after the rogue Night's Watch men kill him and the Old Bear:
Gilly was crying. “Me and the babe. Please. I’ll be your wife, like I
was Craster’s. please, ser crow. He’s a boy, just like Nella said he’d
be. If you don’t take him, they will.”
“They?” said Sam, and the raven cocked its black head and echoed,
“They They They”
“The boy’s brothers,” said the old woman on the left. “Craster’s sons.
The white cold’s rising out there, crow. I can feel it in my bones.
These poor old bones don’t lie. They’ll be here soon, the sons.”
This implies that Craster's sons are somehow turned into White Walkers. Which also presumably is why Craster and Craster's Keep is not attacked by the White Walkers or Wights. I believe Craster says something to the effect that he is a godly man, and does not need to worry about wights.