c. 1200, "the young of a goat," from a Scandinavian source such as Old Norse kið "young goat," from Proto-Germanic *kidjom (source also of Old High German kizzi, German kitze, Danish and Swedish kid), of uncertain origin.
Extended meaning "child" is first recorded as slang 1590s, established in informal usage by 1840s. Applied to skillful young thieves and pugilists at least since 1812. Kid stuff "something easy" is from 1913 (the phrase was in use about that time in reference to vaudeville acts or advertisements featuring children, and to child-oriented features in newspapers).
In clothing, "made of soft leather," as though from the skin of a kid, but commercially often of other skins. Hence kid glove "a glove made of kidskin leather" is from 1680s; sense of "characterized by wearing kid gloves," therefore "dainty, delicate" is from 1856.
"tease playfully," 1839, earlier, in thieves' cant, "to coax, wheedle, hoax" (1811), probably from kid (n.), via notion of "treat as a child, make a kid of." Related: Kidded; kidding. Colloquial interjection no kidding! "that's the truth" is from 1914.