Going with the themes above...
A test of doctrine.
A test of morals.
A test of wild card.
Now, if this is a monastery that regularly has visitors coming and going, they probably can't get too worked-up for these tests. The test of doctrine should be simple, the test of morals a striaght-forward ethical dilemma (think Philosophy 101), and the "wild card" could just be "passport and visa, please".
If it's a more cloistered place where guests are infrequent, then the monks might make the tests way more complicated just 'cause it's the only kind of fun they get. The test of doctrine could be something really obscure (so not something they can research ahead of time), but it just so happens that Brother Drones-On-And-On is in the nearby village giving a sermon later today on that very topic. The PCs get there, and it turns out that Brother Drones-On-And-On is aptly named, and it's a challenge for the PCs to stay awake to get the answer they need.
The test of "morals" could be a scenario they put the PCs through. Perhaps while the PC is on their way back from listing to Brother Drones-On-And-On they run into a (fake) beggar requesting some kind of aid. The aid shouldn't be easy/quick, but something that would delay them from their task for at least a day or something. The "test" is to see if the PCs do the task or continue on their way. Bonus points if you make it so that the PCs have a fair chance of figuring out it's a test and not a person that's actually in trouble.
And the wild card... if the monks are having fun with this, then it should be some petty and tedious task... Go to the silver flame chapel in the nearby town, and carry back a torch lit from the sacred flame... oh, and did we mention it's a super rainy day?
If the monks are having fun, then it's also possible to "fail" all the tasks, but get in anyway. The "guard" is going on about how awful and terrible the PCs are for failing their trials, and then the head monk (or whatever you want to call the rank) shows up saying "what's going on? Are you harassing pilgrims again? For your own boredom, even?!" and so-on.