The word tweezer comes from etwee which describes a small case that people would use to carry small objects (such as toothpicks) with them. Etwee takes its origin from French étui "small case" from the Old French verb estuier, "to hold or keep safe." Over time, the object now known as "tweezers" took on this name because the tool was commonly found in these tiny carrying cases. Eventually, the word "tweeze" was accepted as a verb in the English language.
There is even evidence of Roman shipbuilders pulling nails out of construction with plier-type pincers. How interesting is that!