1650s, powny, from Scottish, apparently from obsolete French poulenet "little foal" (mid-15c.), diminutive of Old French poulain "foal," from Late Latin pullanus "young of an animal," from Latin pullus "young of a horse, fowl, etc." (see foal (n.)) [Skeat's suggestion, still accepted].
German, sensibly, indicates this animal by attaching a diminutive suffix to its word for "horse," which might yield Modern English *horslet. Modern French poney is a 19c. borrowing from English. Meaning "crib of a text as a cheating aid" (1827) and "small liquor glass" (1849) both are from notion of "smallness" (the former also "something one rides"). As the name of a popular dance, it dates from 1963. The U.S.