What does a Prime Hermeticon do?
Absent any specific fluff I assume they administrate the lower ranking Magos of the Mechanicus and assign them tasks in pursuit of the Hermeticon's plans and projects, or the greater goals of the Mechanicus as a whole.
They're the highest ranking of the Magos, so it seems reasonable to assume they govern over the others when they aren't busy working in their preferred field.
I know what you're referring to with "highest ranking of the Magos" but I don't think that's correct. I think it's some kind of specific speciality, like how Magi Biologis specialise in the study of biology or how necromechanics focus on repairing the vehicles and personnel of the Mechanicus.
Codex Adeptus Mechanicus has this for the Prime Hermeticon Warlord trait:
Let none doubt the cold and logistical hatred of the Machine God and its power to invigorate the faithful.
Friendly <FORGE WORLD> INFANTRY units within 6" of your Warlord can re-roll failed hit rolls in the Fight phase.
There's also the Prime Hermeticon in Mechanicus, who gives you missions to hunt down bosses (while being rather enthusiastic about it) and after beating one of them says that cleaning up the normal forces afterwards are outside what a Prime Hermeticon cares about.
I guess their job is to make people more enthusiastic about violence?