I’m with him. My personal house is always in the 60s. Bu that’s personal preference.
If the thermostat is set too low it will develop ice/frost on the coil inside of the unit. The more frost and ice develop the air flow is restricted and the coil becomes a block of ice. Some systems can handle 60 degree temps and some cannot. The coil would need to be VERY CLEAN! Any lint, hair, dirt on the coil will slow air flow and develop ice. In addition the fan speed needs to be set to HIGH when the unit is in cooling mode. You need a large volume of air across the coil to keep frost from forming. This also means keeping the filter clean as well.
One other trick, the temperature on most digital thermostats can be ‘adjusted’ about 3 degrees up or down. When it is 70, you can ‘adjust’ it to say 67. Your HVAC tech can do this for you.
Overall, the unit needs to be in tip top shape and clean to get it down to 60 with out freezing.
If he’s paying the electric bill I’d say let him run it at whatever he wants. If the system is not capable of getting the place cool enough for him he can either a) move out or b) convince you to repair the current system or put in a new system capable of keeping it as cool as he prefers.
If it was me, I would let the tenant run it how they want but definitely not pay for a newer system. Having an HVAC professional check it out isn’t a bad idea. Perhaps your ductwork is leaking or there is another easy fix. You can always find another tenant who won’t care if the AC doesn’t reach the lower 60’s though.
If he runs it too cold, too long, or when the temperatures outside are too low, it can damage the unit. If you are bringing a technician out, ask him what the proper operation procedures are and then pass that on to your tenant. If the system dies because the tenant is abusing it, charge him for the repair.
All depends on what the outside temperature is, AC units are only expected to reduce the inside temp by 25-30 degrees and that is largely dependent on the energy efficiency of the insulation and windows. If it's over 88 degrees outside he may not get what he wants no matter how new or good the AC unit is. A good AC tech can verify this exact info for you.