Hill is topographic English and Scottish naming at its clearest: it marked the household on or beside a hill. When official surnames hardened, “John at the hill” could become John Hill. Because hills are everywhere, the surname arose in many places at once; two Hills need not share a medieval grandfather.
It is a place-type name, not a job and not a personality hint. Occasional other origins exist at the margins of surname dictionaries, but the living-on-a-hill explanation is the one major references lead with. For a printable family page, that plain landscape story is both accurate and easy to tell kids.