1570s, "cotton padding," corrupted from earlier bombace "raw cotton" (1550s), from Old French bombace "cotton, cotton wadding," from Late Latin bombacem, accusative of bombax "cotton, 'linteorum aut aliae quaevis quisquiliae,' " a corruption and transferred use of Latin bombyx "silk," from Greek bombyx "silk, silkworm" (which also came to mean "cotton" in Medieval Greek), from some oriental word, perhaps related to Iranian pambak (modern panba) or Armenian bambok, perhaps ultimately from a PIE root meaning "to twist, wind."
Also from the same source are Swedish bomull, Danish bomuld "cotton," and, via Turkish forms, Modern Greek mpampaki, Rumanian bumbac, Serbo-Croatian pamuk. German baumwolle "cotton" probably is from the Latin word but altered by folk-etymology to look like "tree wool." Polish bawełna, Lithuanian bovelna are partial translations from German.
From stuffing and padding for clothes or upholstery, meaning extended to "pompous, empty speech" (1580s).
Bombast was originally applied to a stuff of soft, loose texture, once used to swell the garment. Fustian was also a kind of cloth of stiff expansive character. These terms are applied to a high, swelling style of writing, full of extravagant sentiments and expressions. Bathos is a word which has the same application, meaning generally the mock heroic--that "depth" into which one falls who overleaps the sublime; the step which one makes in order to pass from the sublime to the ridiculous. [James de Mille, "Elements of Rhetoric," 1878]