Sure, I'd like to know more. I'm not sure if it is a super rare object, though. Instead of being tarnishing, the black substance could be some kind of black washing (i.e., to bring out the details more). The craftsmanship is not of a particularly high quality either. I would expect something to be given to an exam hopeful to be of a higher quality. For instance, here is a gift that was given to one such person in the 18th century. It is made from white jade and zitan wood:

One thing you have to remember is that the exams continued, at least in China, until the early to mid-20th century (it eventually started up in Taiwan and continues today). So even if it was given to a hopeful, it might not be that old. Another thing that tells me that it might not be rare is the fact that it was obviously created in a mold. This means that many of these had to have been made at one time. An artisan is not going to make a mold for something this tiny and only make one. I see that it has a hole on the top, almost like it was meant to be hung from a rope. It’s even possible that this doesn’t have any direct connection to the exams at all. It could just be a wall decoration given to students. It could just be a reminder to students to study hard like those wishing to pass the civil exams.
I think the major question is where exactly did your student purchased it? A high tourist traffic area perhaps? Another important one is how much did they pay for it?