Why is the word "complement" used in set theory?

The first non-mathy definition that Merriam-Webster gives for "complement" is

something that fills up, completes, or makes perfect

You may also note that the other definitions have a similar connotation. Things complement each other if, when put together, they make up something that is complete. For another mathematical example, consider complementary angles, which add up to a "perfect" right angle.

In set theory, a set and its complement form a universal set (i.e. if $X$ is the universe, then $A \cup A^c = X$). Hence the two sets complement each other, in the sense that together, they make a whole.

Addendum: the Oxford English Dictionary (this may require academic access) indicates that "complement" was first used in the mathematical sense in Billingsley's 1570 translation of Euclid's Elements. At that time, it had been in English usage for at least 150 years (the earliest reference in the OED goes back to 1419, as far as I can tell, meaning something to the effect of "finishing or completing something"). The word originally comes from Latin: "complēmentum that which fills up or completes."

Given how you talk about this, I suspect that part of your confusion is that you seem to think that the complement operator applies to some other operator (in your specific example, a union, but whatever other operator would end up there). But, complement is an operator that is applied to sets, not to operators. So, I can just have $A'$ as the complement of set $A$. And in the case of $(A \cup B)'$, the set to which you apply the complement operator happens to be the union of two sets.

So the complement is not, as you write, 'all the things that the original operator is not' (my emphasis) but rather 'all the things that the original set is not' (better put: 'all the things that are not in the original set'). Likewise, in your case, the complement does not 'complete' the operation of union, but simply 'completes' a set ... that again just happens to be a union of two sets.

"X complements Y" in colloquial English means basically "X has what Y lacks". This is exactly what the complement is in set theory, except that the complement of $A$ also has none of what $A$ has.

That said, my suspicion is that this term actually originates in mathematical French and was borrowed directly from mathematical French into mathematical English.