literally means the produce of the bombyx (Middle Latin bombax, Greek bombux), and applied to cotton-wool used for padding. The head of the cotton plant was called “bombast” or “bombace” in the sixteenth century. Bombast was much used in the reign of Henry VIII. for padding, and hence inflated language was so called.
We have received your letters full of love, ... And in our maiden council rated them ... As bombast and as lining to the time.
Shakespeare: Love's Labour's Lost, v. 2.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894