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.
From the get go Larian has been pretty transparent that they created a different kind of a tadpole for their needs.
Others endow you with psionic abilities to push and pull enemies like ragdolls in battle.
So... push?
I don't want to get my hopes up, but in the early days Larian explained over the top jump and push of our characters as an influence of the tadpole. Wouldn't it be cool, if those weren't as overdone by default, and player could sacrifice to the tadpole to gain those powers? I doubt that will happen... but I think that would be cool.
