Lets say I have my palm flat with a book resting on top of it, and I have my feet on the ground. I extend my arm so that now it's kind of difficult to keep the book up. Why doesn't my hand just produce normal force on the book, cancelling out the force of gravity, and costing me no effort whatsoever?
Why doesn't my hand just produce normal force on the book, cancelling out the force of gravity
It does. Since the book doesn't accelerate downwards, another force is compensating for the weight. That is the normal force from your hand.
and costing me no effort whatsoever?
You are right that the normal force does not require energy to withstand the weight. Like the normal force from a table. The only reason you feel tired is because of the work of muscles in your body. Their stretching and contractions are energy consuming. It's not the normal force that makes you tired, but rather what the body does to produce that normal force.
And now to your headline question:
Why can't I harness normal force?
Essentially, you can harness work while something is moving or exchanging heat. As said above, the normal force of any object does not require nor absorb energy to keep itself up. You could ask the same question about the weight from gravity. Why can't we harness energy from that? Because something must move or exchange heat for us to extract work. A book lying on the table is of no power-use in that state.
The force it needs to hold the book in its position is the same in both cases. This forces is caused by gravity.
The book doesn't move (accelerate) which means there must be some other force, working against gravity. The sum of all forces is 0.
I extend my arm
And this is what makes the difference here. You are not just talking about your hand, but your entire arm. Newton is not enough. You also need Archimedes' torque to explain what's going on.
It's not just the sum of all forces that has to be 0, but additionally, the sum of all torques has to be 0. A torque is a force on a lever. That lever is your arm.
If you extend your arm, you extend the lever. By extending the lever, the torque of the book increases (remember: the force remains the same). You have to compensate that bigger torque, because the sum of all torques has to be 0 if the book should not move. That's what makes holding the book with an extended arm more difficult.
Name has a really great answer, but I'm gonna try to clarify it a little. If you hold the book, its very unlikely that our arm is perfectly at 90 degrees, perpendicular to the ground. That means you will have to counteract the book's torque. But even if you held your arm perfectly at 90 degrees you would still get tired. It's not like the book is perfectly above your center of gravity, and that is also causing some torque. Also, think about all the joints in your body from the top of your arm to your foot, they are not all perfectly aligned so the muscles need to do work so that you don't collapse.
I haven't seen this mentioned yet: this normal force is in the context of classical mechanics (high-school or early university physics). That theory only deals with (perfectly) rigid bodies. So you can put a weight on a hypothetically perfectly rigid table and the table does not have to do any work to support it. The table doesn't compress even the tiniest bit. So in this model, work only happens when there's a displacement of the body.
Your body is not perfectly rigid. You could model it classically as rigid parts, connected by springs. In that case you'd see that, yes, you do need to input energy to the springs (muscles) in order to hold a pose when there's a weight on your hand. This more closely matches what you "feel". Holding up a weight takes effort, even if you're not moving the weight.
The core of the problem is modelling a non-rigid body as a rigid body. Same as spherical cows and point-mass planets.