1650s, formerly also elastick, coined in French (1650s) as a scientific term to describe gases, "having the property of recovering its former volume after compression," from Modern Latin elasticus, from Greek elastos "ductile, flexible," related to elaunein "to strike, beat out," which is of uncertain origin; according to Watkins from an extended form of the PIE base *ele- "to go."
Applied to solids from 1670s, "having the power of returning to the form from which it is bent, etc., as soon as the applied force is removed." Figurative use by 1859. The noun meaning "piece of elastic material," originally a cord or string woven with rubber, is from 1847, American English.
1859, from Elasmobranchii, class of fishes that includes sharks and rays, from combining form of Greek elasmos "metal plate," from elan "to strike" (see elastic) + brankhia "gill." So called from their plate-like gills.