c.1300, "implement for throwing stones," from an unidentified continental source, e.g. Middle Low German slinge "sling," from Proto-Germanic *slenganan (cf. Old High German slingan, German schlingen "to swing to and fro, wind, twist;" Old English slingan "to creep, twist;" Old Frisian slinge, Old High German slinga, German Schlinge "sling;" Middle Swedish slonga "noose, knot, snare").
The notion probably is of a sling being twisted and twirled before it is thrown. Sense of "loop for lifting or carrying heavy objects" first recorded early 14c. Meaning "piece of cloth tied around the neck to support an injured arm" is first attested 1720.
"sweet flavored liquor drink," 1807, American English, of unknown origin; perhaps literally "to throw back" a drink, or from German schlingen "to swallow."