Zen traditions say that in looking for the mind, there is nothing to find.
It's not just Zen traditions. Tibetan Buddhism have the same exercise, usually placed at beginner-to-intermediate level, right after the teaching on six realms and karma and before "real" meditation.
The idea is that most beginners have this unexamined idea that "I" or "mind" is some thing inside our heads. Trying to examine that thing, locate it in space, and describe its shape and characteristics is the first attempt at introspection that the student takes on. Eventually, the student is supposed to realize that what they call "I think" are just thoughts automatically coming from memory in response to a stimulus, and learn to disidentify from the thoughts.
And, if the body can be found, do we identify with it?
I appreciate your sarcasm. It's not that "not finding"
makes it easier to disidentify, it's seeing that mind is not "one thing" and that "I" is just something we literally imagine.
Huike said to Bodhidharma, “My mind is anxious. Please pacify it.” To which Bodhidharma replied, “Bring me your mind, and I will pacify it.” Huike said, “Although I’ve sought it, I cannot find it.” Bodhidharma then said, “There, I have pacified your mind.”
The Huike story, although seemingly about the same topic, is actually a reference to another principle of Buddhism, the principle of "immediacy".
The problem with most students is that their minds focus on Buddhist theory while ignoring their immediate state of mind, with its deep-lying conflict. Even the quest for Nirvana is a state of conflict, as we identify "this" as "wrong" and Nirvana as right. No amount of theory helps until the student admits this conflict in their own mind. So turning the student's mind to the present moment, to help it see its own conflict is THE core principle of Zen. This is why Zen is called "the school of direct pointing, beyond theory" (or something like that)
That's why Bodhidharma says, "There, I have pacified your mind." - this last phrase is the "pointing directly at mind". At that moment Huike must have turned his attention to his own mind, and saw how grasping (the frustrating and hopeless act of trying to pinpoint the mind) was the very act that created his suffering, and how letting go of that grasping was letting go of that specific instance of suffering -- which was the stream-entry moment for him.
The same principle is at work behind koans, when the act of letting go of the impossible problem leads to realization of the Third Noble Truth.