"laced underbodice," c. 1600, from plural of stay (n.1). Also compare bodice.
Entries linking to stays
stay n.1
"support, prop, brace," 1510s, from French estaie "piece of wood used as a support," Old French estaie "prop, support," perhaps from Frankish *staka "support" or some other Germanic word, from Proto-Germanic *stagaz (source also of Middle Dutch stake "stick," Old English steli "steel," stæg "rope used to support a mast"), from PIE *stak- "to stand, place" (see stay (n.2)). In some uses from stay (v.2).
bodice n.
1560s, oddly spelled plural of body, originally the name of a tight-fitting Elizabethan inner stays or corset, laced in front, covering the torso, worn by women and sometimes men; plural because the body came in two parts which fastened in the middle. For the spelling, compare deuce. In modern use, an outer laced garment covering the waist and bust worn by women, often as an ornament.
Bodice-ripper for "racy romance novel" is from 1981. Related: Bodiced.