During that edition, the backstory of the God-Emperor of Mankind was that he had been born some fifty thousand years before the game took place. Back then, humanity was being led and advised by psychically-powerful individuals known as Shamans. They noticed that, as they died, they weren't all reincarnating properly, because the Warp was getting fouled up by Chaos and the nascent Chaos Gods were trying to eat their souls. They decided to pool their power by committing ritual suicide and reincarnating into one tremendously powerful individual, who eventually became the God-Emperor.
The Star Child hypothesis states that, like the Shamans, the Emperor is capable of reincarnation, but because he wasn't quite killed by Horus, his soul is still anchored to his wrecked shell of a body. Therefore, in order for the Emperor to be reborn, his body will have to die completely.
This would cause some serious trouble for the Imperium, as they would be bereft of the Astronomican for however long it took for the Emperor to get used to his new body and make his way back to Terra. In fact, it was suggested in the Third Edition rulebook that at least one of the Star Child cults was actually being used by Tzeentch for this purpose, which makes the entire theory all the more doubtful.
Currently, it's uncertain if the original origin story is even real, given a large amount of inconsistencies with the Horus Heresy novels and the introduction of the Perpetuals retconning the original backstory of the Emperor entirely. As of 8e, it can be safely assumed that it was just another piece of Rogue Trader weirdness that didn't survive the changing editions.
The concept is not entirely gone, however. Xenology references an Eldar artfifact depicting a child enveloped in sunlight, which, considering its origins, is most likely Ynnead. Unlike the Star Child, Ynnead is born of Eldar souls, but can still be considered to be descended from the original idea.