1650s, "melancholy," from French chagrin "melancholy, anxiety, vexation" (14c.), from Old North French chagreiner or Angevin dialect chagraigner "sadden," which is of unknown origin, perhaps [Gamillscheg] from Old French graignier "grieve over, be angry," from graigne "sadness, resentment, grief, vexation," from graim "sorrowful," which is perhaps from a Germanic source (compare Old High German gram "angry, fierce").
But OED and other sources trace it to an identical Old French word, borrowed into English phonetically as shagreen, meaning "rough skin or hide" (the connecting notion being "roughness, harshness"), which is itself of uncertain origin. The modern sense of "feeling of irritation from disappointment, mortification or mental pain from the failure of aims or plans" is from 1716.
The town of Chagrin Falls in northeastern Ohio, U.S., was founded 1837 and named for the nearby falls of the Chagrin River. The source of the river name is uncertain; it might be a corruption of Seguin, the name of a Frenchman who is said to have established a trading post nearby in the 1740s.