Last names passing from father to children is a comparatively recent phenomenon. This is obvious upon reflection - so many surnames today are references to the life or activities of a long-ago person, like an occupation, city or other location, life event or even physical characteristic. Same as today, last names were used to distinguish all the (say) Johns in a community, only they could describe the individual themselves and not necessarily their family.

Just what gained a person their last name isn't always so easy to know. My last name (Pace) seems obviously to refer to the Latin word for "peace", but it could be a reference to Easter (Pascha), or to Pacy-sur-Eure, in Normandy.

Same with Griffin. Maybe in many cases it was just a place name reference. But perhaps, in some cases some long-ago fellow was a brave and resourceful fighter, and was renowned for having the bravery of a Griffin (or perhaps he was a coward and the name was bestowed by his neighbors ironically, that could have happened - here's the best example I can think of for that). Or maybe the individual had a large, beak-like nose and gained the name that way. Who knows.

Last names are a continual fascination for me - if you think about it, your last name represents a single person from your family tree who lived 200-1000 years ago, who won a giant generation-by-generation single elimination tournament. Who they were, where they lived, what they were like you have no idea, but you know their name and you go by it too.