Origin and history of slung

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]