It turns out the true etymology of ferret is a bit elusive. It’s thought to be derived from Latin ferre “to bear” (ex: fertile “able to bear”), related to Greek phor “to bear.” Ultimately from Proto-Indo-European bher “to bear, carry” (see Greek words like euphoria “to bear wellness,” metaphor “to bear across or beyond,” phosphorus “light-bearer”).
So in that sense, the ferret was used to hunt and bring out rabbits and rats; hence to ferret out.

Ferret is also suggested to be derived from Latin fur “thief” and furritus meaning “little thief,” a likely reference to the ferret tendency to secret away small items. Furtive is of the same etymology; literally “thief-like.” From Old French via the l/r shift, we have felon, felonius, felony.

While looking up “ferret,” I was amused that a group of ferrets is called a business, so essentially a business of little thieves.
I’ll also leave this Bojack Horseman joke here.😆

Source: John T. Shipley’s The Origins of English Words: A Discursive Dictionary of Indo-European Roots (1984)