Thank you everyone for the information! I figured out the problem- the bobbin tension was absurdly high! The sticky "grinding" feeling was the machine struggling to pull the thread from the bobbin, and the "click" was when the thread "released", as it were.
I'm surprised it didn't break! I had to turn the tension screw on the bobbin case almost a full turn to loosen it up so somewhere resembling "correct". I don't have any way to measure bobbin tension, but I was able to get it approximately right by comparing how hard it was to tug the bobbin thread through my bobbin case, and through the bobbin case from my mom's machine (same model). Then I had to re-calibrate the top tension, which wasn't too hard to figure out. Now she sews beautifully, no clicking or grinding or any worrying noises!