slingshot n.

c. 1300, "hand-held implement for throwing stones" (consisting of a strap and two strings), from an unidentified continental Germanic source (such as Middle Low German slinge "a sling") from verbs meaning "swing to and fro, wind, twist" (see sling (v.1)).

The notion probably is of being twisted and twirled before it is released. The piece of stone or metal hurled from it is a sling-stone (late 14c.). The older English word for this ancient weapon was lithere (Old English liþere, related to leather).

Some of the sense developments might have been influenced by Low German cognates. The sense of "loop for lifting or carrying heavy objects" is recorded by early 14c. As "leather shoulder strap for a rifle, etc.," by 1711. The meaning "piece of cloth tied in a hanging loop around the neck to support an injured arm" is attested by 1720. A Middle English medical word for "sling or supporting loop used in treating dislocations" was stremb (early 15c., Chauliac, who also has suspensorie), from Medieval Latin stremba. 

The Greeks, Romans, and Carthaginians had bodies of slingers attached to their armies, recruited especially from the inhabitants of the Balearic Isles. The use of the sling continued among European armies to the sixteenth century, at which time it was employed to hurl grenades. [Century Dictionary]

Middle English shot "a missile, arrow, dart" (senses now archaic or obsolete); "a swift movement, a gushing out," from Old English scot, sceot "a shot, a shooting, an act of shooting; that which is discharged in shooting, what is shot forth; darting, rapid motion."

This is from Proto-Germanic *skutan (source also of Old Norse skutr, Old Frisian skete, Middle Dutch scote, German Schuß "a shot"), from PIE root *skeud- "to shoot, chase, throw." The Old English noun is related to sceotan "to shoot." The meaning "discharge of a bow, missile," also is from related Old English gesceot.

The noun was extended to other projectiles (balls, bullets) by mid-15c. Especially "lead in small pellets, a small ball or pellet," a number of which are combined in one charge, which is attested by 1770 (shortened from earlier small shot, 1727).

The general sense of "an attempt to hit with a projectile" is by 1650s. Extended to sports (hockey, basketball, etc.) by 1772, originally in curling. It is attested by early 15c. as "range or distance of a missile in flight," hence "range" in general (c. 1600), as in earshot. 

Another original meaning, "payment" (perhaps literally "money thrown down") is preserved in scot-free; also see scot (n.). The notion of "throwing down" might have led to the meaning "a drink," first attested 1670s; the more precise meaning "small drink of straight liquor" is by 1928.

The sense of "hypodermic injection" is attested from 1904; the figurative phrase shot in the arm "stimulant" is by 1922. The broad meaning "a try, an attempt" is by 1756; the sense of "remark meant to wound" is by 1841. The meaning "an expert in shooting with a firearm" is from 1780; the sense of "a rocket flight" is by 1934. The camera-view sense is by 1958.

To call the shots "control events, make decisions" is American English, 1922, perhaps from sport shooting. Shot in the dark "uninformed guess, random attempt" is by 1885. Big shot "important person" is from 1861. 

By the rude bridge that arched the flood,

   Their flag to April’s breeze unfurled,

Here once the embattled farmers stood

   And fired the shot heard round the world.

[Emerson, from "Concord Hymn"]