Old English þing "meeting, assembly," later "entity, being, matter" (subject of deliberation in an assembly), also "act, deed, event, material object, body, being," from Proto-Germanic *thengan "appointed time" (cf. Old Frisian thing "assembly, council, suit, matter, thing," Middle Dutch dinc "court-day, suit, plea, concern, affair, thing," Dutch ding "thing," Old High German ding "public assembly for judgment and business, lawsuit," German ding "affair, matter, thing," Old Norse þing "public assembly"). Some suggest an ultimate connection to PIE root *ten- "stretch," perhaps on notion of "stretch of time for a meeting or assembly."
For sense evolution, cf. French chose, Spanish cosa "thing," from Latin causa "judicial process, lawsuit, case;" Latin res "affair, thing," also "case at law, cause." Old sense is preserved in second element of hustings and in Icelandic Althing, the nation's general assembly.
Used colloquially since c.1600 to indicate things the speaker can't name at the moment, often with various meaningless suffixes, e.g. thingumbob (1751), thingamajig (1824). Southern U.S. pronunciation thang attested from 1937. The thing "what's stylish or fashionable" is recorded from 1762. Phrase do your thing "follow your particular predilection," though associated with hippie-speak of 1960s is attested from 1841.
To follow one's special inclinations, esp despite disapproval; to act independently or alone: Doing your own thing may be all right in prescribed doses (1840+)