Question: I can't remember when and where I had this discussion, but I remember being corrected when I was speaking by a stranger saying that it is never correct to say *give me half of this*; instead, the grammatically correct phrase would be *give me one half of this*. I've never been a pro at where numbers fit in with the English language, so maybe someone here could shed some light on this.

Answer: 
There is no need to say "one half" there. "Give me half of that" is sufficient.
If you are discussing arithmetic, it is always correct to stipulate a numerator and a denominator. "Give me half an apple" is fine, but "what is half plus a third?" is incorrect and should be "what is one-half plus one-third?"