C172 flight instructor with student reported an engine malfunction during cruise on a night training flight. The instructor took control; diverted to an alternate airport; and landed safely.

Narrative

I was performing a night 100NM XC flight with a student from ZZZ to ZZZ1 to ZZZ for his commercial license requirements; and to familiarize him with our TAA 172s avidyne panel. Before flight; we satisfied all pre flight information requirements; and decided to select 8500 as our cruise altitude; a higher altitude than normal to give us more options if we were to experience any type of engine problems. The departure and climbout were standard and uneventful. Most of our attention was on the oil/chts in the climb; due to the high temperatures; making sure to execute a very gradual cruise climb. We leveled out at 8500 ft; set the mixture; and programmed the autopilot to navigate direct and hold altitude. approximately 10 minutes into cruise; the engine completely stopped for three seconds. I immediately disengaged autopilot; and told my student i had the controls. The engine started coming back; but would lose power in cycles of 2-3 seconds continuously. We immediately began verifying tanks were on both; we had the proper mixture; mags were good; primer was locked; carb heat was left on; and all engine pressured and temperatures were normal. I immediately [requested priority handling] and obtained vectors to the nearest airport ZZZ2 from ZZZ center. We had absolutely no idea what was causing the issue. I located the airport; and spiraled down overtop of it; assuming my engine could fully fail at any moment.we noticed as we descended the engine seemed to be holding power when we tested the throttle; but didn't want to count on it. we entered the pattern for rwy XX; and landed slightly long without incident. we shut down the aircraft and inspected the plane thoroughly; checking fuel for contamination again; making sure we had oil; and noticed no issues. After the situation was over; it was brought to my attention that there was a section in the 172F; that recommends switching from the BOTH fuel position; to either the L or R tanks after leveling cruise on hot days above 5000 ft MSL; to prevent vapor lock issues. Having over 50 hours in 172s; and 700 hours in 150s; i had never experienced this issue; or had any problem cruising on the BOTH position for the entire flight. The last time I had reviewed this POH was several months earlier; and this odd tank switching procedure was not on my mind. If I had simply switched tank positions in cruise flight; I could have avoided this [priority handling] situation. Once again the vapor lock issue never occurred to me after the engine stopped running; and I never switched the tanks off the both position; as every [priority handling] checklist says to keep tanks on BOTH. I'm sure fatigue had also played a slight role as well; since I was a little tired having woken up at XA:00 am that day; although I had extremely good sleep. What I've learned is that its very important to stay familiar with all the recommended procedures for the specific make and model you fly; and not to neglect switching tanks in cruise; even if it seems like an unnecessary procedure. Ive learned some valuable lessons from this incident; and will make sure my students never make the same mistakes; and are very familiar with the procedures laid out in the POHs of the aircraft they fly.