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.
I had this situation recently. It was so embarrassing when the saleslady told her co-workers about what I said. 
I asked her to give me one half kilo (1/2 kilo) of prawns and 3 pieces of fish. She ended up giving me one and one half kilo of prawns. When she asked for the payments and I asked the amount, i freaked out. I was not expecting that she actually gave me 1 1/2 kilo of prawns. 
I told her I asked for one half not one and one half kilo. She got angry and her co-workers started to laugh. They were all telling me I was wrong. They said it's half not one half.
In my understanding, "one half" and "half" are both correct it depends on how you use it with other supporting words.