Middle English strau, from Old English streaw (rare) "dried stems or stalks of certain grains after threshing," apparently literally "that which is scattered or strewn," related to streowian (see strew).
This is from Proto-Germanic *straw- "that which is scattered" (source also of Old Norse stra, Danish straa, Swedish strå, Old Saxon stro, Old Frisian stre, Old Dutch, Old High German stro, Dutch stroo, German Stroh "straw"), from PIE root *stere- "to spread." The notion perhaps is of dried grain stalks strewn on a floor as carpeting or bedding. As a type of what is trifling or unimportant, attested from late 13c.
The meaning "hollow tube through which a drink is sucked" is recorded from 1851. As an adjective, "made of straw," mid-15c.; hence "false, sham," as in straw bid (by 1889).
Straw poll "vote taken without previous notice or at a casual gathering" is from 1932; earlier straw vote (1866). Straw hat, one made of woven or plaited straw, is attested mid-15c. (strawen hattis).
To clutch (or grasp or catch) at straws (1748) is what a drowning man proverbially would do. The last straw (1836 apart from the full phrase) is from the proverbial "it is the last straw that breaks the camel's back" (or, less often, the mare's, the horse's, or the elephant's), an image in use in English by 1755.
Let it not, however, be inferred that taxation cannot be pushed too far : it is, as the Oriental proverb says, the last straw that overloads the camel ; a small addition, if ill-timed, may overturn the whole. [The Scots Magazine, April 1799]
To draw straws as a means of deciding something is recorded from 1779 (the custom probably is older). It also meant, in reference to the eyes, "give indications of sleepiness" (1690s). In the straw, of a woman, "giving birth" is from 1660s.