For the reference, the test scene:
Kay and Zed are observing the participants take the test, and discussing them. The whole thing is absurd:
- the chairs are good only to crouch inside, but not to do any work (like taking a simple test) in them;
- the test are sealed in a way that makes them hard to open (the paper rips easily, the pencil breaks when Jay tries to use it to pry the seal open,...).
The table seems to be the only solution, but it's not offered as such.
So, I believe this is a test of how well they manage themselves in absurd situations (which they get plenty of in the field, as we see later on in all three movies), and are they capable of thinking outside of the box.
Then there is a test at the shooting range. Obviously, "the best of the best" know how to shoot, so their ability to just aim and shoot is not much of an issue. However, when the shooting is done, Zed comes in and asks Edwards "May I ask why you felt little Tiffany deserved to die?" (at 3:42 in the linked clip). This is Jay's answer:
Well, first I was gonna pop this guy hanging from the street light, and I realized, y'know, he's just working out. I mean, how would I feel if somebody come runnin' in the gym and bust me in my ass while I'm on the treadmill? Then I saw this snarling beast guy, and I noticed he had a tissue in his hand, and I'm realizing, y'know, he's not snarling, he's sneezing. Y'know, ain't no real threat there. Then I saw little Tiffany. I'm thinking, y'know, eight-year-old white girl, middle of the ghetto, bunch of monsters, this time of night with quantum physics books? She about to start some shit, Zed. She's about eight years old, those books are WAY too advanced for her. If you ask me, I'd say she's up to something.
To this, Zed smiles contentedly. So, the point in all of the tests (except for the "eye exam" ;-)) is to see how well do the participants handle themselves, and are they capable of thinking critically and out of the box.