Learn English – the origin of the phrase “I rest the case”

I believe this is a variant of "hello the camp", which is used as your father describes: when you're about to enter someone's encampment, as a kind of warning/greeting in lieu of knocking on the nonexistent door. (There's an elided "to" in there: the full/grammatical form would be "hello to the camp".)

I imagine "hello the fire" could be used if there isn't an encampment to speak of, just a fire with guys sitting around it.

The phrase is Irish in origin but now very rarely used in Ireland (except as a sterotypical "Irishism"). It simply means "the best of the morning to you" - perhaps from the idea of unhomogenised milk, where the cream rises to the top. An appropriate response might be a simple "thank you" although the traditional response would be "And the rest of the day to yourself."

Terrible attempts at Irish accents, dancing a jig and leprechaun costumes are entirely optional while saying this.