Original Punc Rock
To punctuate, from the Latin punctuare, means to mark with points or dots. It comes from *peuk, the Proto-Indo-European root meaning to prick. Punctuation shares ancestry with pounce and punch, along with pungent and a bunch of other excellent words. To punctuate writing means to punch it up with points or dots that indicate intent. There a bunch of fun recent punctuation names like octothorp and sextile, but where did the basic set of punctuation come from? As you might guess, mainly ancient Greece.
The full stop is a point (also from *peuk). Latin gives us periodus, or complete sentence, alluding to a cycle that occurs between one full stop and another. A period is a going around, from the Greek peri- (around) and hodos (journey). A sentence, coming from the Latin sententia, is a thought. When you end sentences with periods, the effect is a series of cycles, marked by periods, each containing (and stating) a thought.
From time to time in this series of little goings around, you might need to separate some pieces from others, as if striking them with a hatchet. It could be that the word comma comes from the PIE root *kop—which might also be the root of the word hatchet. In any case, the ancient Greeks used komma to mean both piece that is cut off and clause in a sentence. They also used kōlon to mean a clause, but it was more connected than the komma—a kōlon is more like a part of a verse, more literally the limb of a person (or a tree). The combination of komma and kōlon—what we think of as a semicolon that joins two independent clauses—is what the ancient Greeks used as a question mark.
Those Greeks had a lot of vivid ideas for the little marks that accompany our goings around. An apostrophe is a turning away. A parenthesis is a putting in beside. An asterisk is nothing more than a little star. And a hyphen is togetherness personified.