"mark indicating an omitted letter," 1580s, from Middle French apostrophe, from Late Latin apostrophus, from Greek apostrophos (prosoidia) "(the accent of) turning away," thus, a mark showing where a letter has been omitted, from apostrephein "avert, turn away," from apo "off, away from" (see apo-) + strephein "to turn" (from PIE root *streb(h)- "to wind, turn").
In English, the mark often represents loss of -e- in -es, possessive ending. By 18c. it was being extended to all possessives, whether they ever had an -e- or not.