In my home the AC system uses a Honeywell TH8110U Touch Screen
Programmable Thermostat. It works fine, but it making a clicking sound
each time it turns on (and off) which is very annoying.

I called Honeywell support for home owners and the support person
explained to me that the clicking sound was normal and almost all of
their programmable thermostats by Honeywell have that click because of
the relay used. He said of those that are programmable that don't have
a click are the Honeywell Magic Stat (models T8112, T8132, T8131), but
they don't have a touch screen and can't be used to switch between
heating and cooling. For now, the heating has its on old analog(?)
thermostat, so that isn't a real concern. I guess worse case, next
winter I will have the one for the heating replaced if it can't be on
the same thermostat as the AC.

Has anyone experiened this clicking problem with the Honeywell
programmable thermostats and replaced it with one of these Magic Stat
models? If so, did this fix the problem? Also, if anyone is using one
of these Magic Stat models does it trully make no click, no noise at
all? Thanks!