I was looking at an exercise this morning which I was able to reduce to showing that the nilradical is the the intersection of the prime ideals in a ring -- a fact I remembered was true, but which I tried for a while to prove without success. Bucking my usual tendency to let something like that ruin the rest of my day for research, I dug out my copy of Atiyah and MacDonald and looked up the answer. (The idea is to assume that some non-nilpotent element f lies in every prime ideal, apply Zorn's lemma to the ideals which contain no power of f ordered by inclusion, and then show that the upper bound is prime.)
My reaction to this was something along the lines of, "Ah, I never would have got that, because I never would have tried using Zorn's lemma!" Upon further reflection, I realized that this indicated a serious weakness in my ability to do commutative algebra.
I'm perfectly comfortable using Zorn's lemma for something like showing that an arbitrary vector space has a basis, but when I look at a question like this I'm just not seeing the connection. I know that this doesn't really have a definite answer, but I was hoping that someone would be able to point out some kind of connection that would improve my intuition for when Zorn's lemma might be effective.
EDIT: Thanks to everyone for all the answers. They are all helpful and I had a hard time choosing!