The moon children are the Happy Mask Salesman's Terminan counterparts. You encounter him in the beginning of Majora's Mask where he explains he's traveled to Termina in search of an ancient mask.
Ha, have you guys ever examined the Happy Mask Salesman as a metaphor for Shigeru Miyamoto? He is said to have been designed with Miyamoto's likeness in mind, hence his particular smile. Because Miyamoto basically rushed Aonuma and his team to make a full Zelda game in a year (1999) as a response to Ocarina of Time's impeccable success, the dev team was stressed to deliver. As a metaphor for Miyamoto's demanding instruction, they decided to take the Happy Mask Salesman—as based on Miyamoto—and portray him in dim light, as a slightly sinister bundle of smiles, happily willing to use you for his endeavors... as if Miyamoto pushed the team to find his prize, a prize cursed by selfish whims. Majora's Mask.
Edit:
Black eyes it seems.
I believe I read your Miyamoto theory somewhere else before and I agree that they at least share a resemblance in face and character. But I don't necessarily believe the moon children are the salesman's counterparts, mostly because there's five of them, they don't wear his clothes, and we never learn for certain what their faces are. There are no other examples of character counterparts beign so radically different. But I think it's supposed to be that way, if you'll entertain this sheikah theory.
Just based on the Zelda Wiki article I've been reading, the three identifying characteristics of the Sheikah are red-eyes, pointy ears, and magical powers. He's got the ears, and he hints at having magical powers. He knows the Song of Healing, which is magical. At the end of the game he just disappears, which is also pretty magical. He's an expert on magical mask stuff. I think it's safe to say that at the very least, he's far from an ordinary man.
The Sheikah are also known for their affiliation with the shadows, and the for the entire game the Salesman waits in Clock Tower, in the dark, behind the scenes. Also, in the beginning to MM he admits to have been following Link, though Link never knew it.
The most telling part for me is the fact that in OoT, he has the Mask of Truth, which is one of three ancient Sheikah truth items. Not only does he have it, he knows what it is and what it does. The two other truth items, the Lens of Truth and the Stone/Shard of Agony, were hidden in Kakariko Village, which is riddled with Sheikah history.
If he is a Sheikah, then we have to consider what his motive is. He says that he has to leave Termina in three days, but we should take that statement with a grain of salt. Remember that time does not move inside the clock tower (another possibly reason to deduce that he has magical powers), so while Link is going through the days and resetting time over and over again, the salesman has been standing there the whole time. In actual time, he would have been standing there for way more than three days. We can see that the "I have somewhere to go" statement is a big lie. and he has to know it because he's stays in the one place where time doesn't matter throughout the whole game.
However, we do know that he enjoys using masks to make people happy. He does it in OoT and MM as well. Tie this concept in with the idea that MM was a spiritual journey for Link and we have something interesting. Perhaps the Salesman was pushing Link into this journey for Link's own sake. Perhaps all of the events of the entire game were orchestrated by the Salesman to help Link. There's some evidence for this as well:
1. Masks are of huge importance to the world of Termina, and they are also, of course, the Salesman's specialty.
2. If you allow the moon to crash, the Salesman is the one who saves you.
3. In the moon the moon children share a resemblance to the Salesman, yet they wear masks that represent all the evils you've been fighting. One could say that in every instance, he is the "man-behind-the-mask." Who's behind the poision at the swamps? The frost in the mountains? The storms at sea? The cursed lands? The moon's fall? You would think that the skull kid would be the answer, but we learn that the skull kid is a puppet, and in the very end the people behind all those respective masks seem to be the salesman himself.
So what if the Salesman was the mastermind behind the entire journey? What becomes of Termina? I think it's possible that the entire land and people were illusions, things the Salesman created to influence Link's spiritual journey. That would explain why there are so many character counterparts, as well as a lot of nonsensical anomalies like a moon with a giant, angry face that has a meadow with a tree inside of it. There is also the fact that the one time the owl is spotted, he mentions that Termina is a land that is destined to fade away. But I'm not very big on this theory.
Lets go back to the idea that he orchestrates the entire journey. Why would that be, and what does it have to do with him being a Sheikah? Remember what he tells Link at the end of the game? He tells him to go home, back to Hyrule. Why Hyrule? Why not tell Link to go off and become his own person or something like that? That statement only makes sense is the Salesman is a Sheikah, because the Sheikah are the servants of Hyrule. Perhaps when the Hero of Time left the land, the Sheikah decided that he needed to be convinced to come back. Maybe Hyrule still needed (or wanted) him to be around. Considering that in this branch of the timeline Ganon is still around, it's possible.
If only he had red eyes, I'd be a lot more sure of this theory.