Why do we use the terminology of Type I/Type II when describing errors in hypothesis testing? I was teaching someone else about type I and type II errors, and they asked the question of why errors are called type I and type II errors. It seems to me that we could call type I errors a "False Rejection" and type II errors a "False Failure to Reject" and thus eliminate the need for students to learn mnemonics to remember which errors are of which type.
Is there any utility to using the Type I/Type II error terminology?
It appears that there wasn't any clear reason for adopting the terms originally

It's really terrible nomenclature that has led to a century of misunderstandings.  The main problem is attachment of the word "error" to the probabilities being calculated, which give a false impression that it's a worthwhile exercise.  Many people act as if type I probability (which I call type I assertion probability $\alpha$) tells you something about the probability you'll make a mistake, e.g., be wrong in concluding an effect exists.  It does nothing like that.  Instead it tells you the chance of asserting an effect when any assertion of an effect is by definition wrong.  It's not an error probability.  More is here which discusses the difference between $\alpha$ and decision errors.
To your original question, type I and type II are convenient shorthand (as long as we drop 'error') but I personally would call them assertion probabilities when, respectively, there isn't or there is a real effect.
