As a student, is it appropriate to use my university's letterhead for a letter I write, to give it extra weight?
Do not use any professional letterhead when you are not speaking "on behalf" of the entity it comes from. 

Using your status, like explaining who you are or whatever, is fine. However, the sole fact of using entity's letter head, implicitly involves your employer and can only ends in a bad situation. 

If you are writing to another department of the same university, like in case of an internal matter, this can be deemed acceptable, but otherwise I would refrain from doing it.