I have never heard of "half illihids" in D&D...ever. Illithids conduct all sorts of mad experiments with tadpoles sticking into every creature they can resulting in all kind of abominations like mind flayer dragons, elder brain dragons, etc. But generally speaking I've only ever heard of two possible results: a) ceremorphosis completes and some new mind-flayer or ulitharid or some other abomination is created depending on the type of creature the tadpole was inserted into OR b) ceremorphosis fails and both the host and tadpole die.
So I haven't the slightest idea what a half-illithid would be or how it would work in relation to elder brains and mind flayer colonies. But it sounds like fun.
Sounds like the aberrant sorcerer origin: You were implanted with a mind flayer tadpole, but the ceremorphosis never completed. And now its psionic power is yours. When you use it, your flesh shines with a strange mucus.
Combined with a bit of Netherese meddling (with a precedent in Ioulaum's transformation into an elder brain lich while preserving his mind) and a splash of quasi-divine intervention, and you've got yourself a potential for half-illithids.
After all, the Netherese Ioulaum both created the first alhoons and illithiliches, and also found a way to turn his lich form into an elder brain.
