What can students do when they apply for a patent in the name of the school?

If you apply in your own name, you are the patentee, and you can control the patent right through transfer or licensing, but you may have to pay the patent application fee, annual fee and agency fee yourself.

Apply in the name of the school, the school is the patentee, you are just the inventor, and you have no right to dispose of the patent, but the school should give you some rewards.

It doesn't matter if it's a small invention. If it's a big invention, I advise you not to. When someone else buys the patent of your invention, they talk to the client, which has nothing to do with you.