Try any route to get out of a desperate situation, no matter how unlikely it is to succeed. A straw was chosen as the height of futility as a means of rescue. Being, as it was, a flimsy and virtually valueless waste product, it was often used as a synonym for the most unimportant and trifling of objects.
It comes from a proverb in Thomas More’s “Dialogue of Comfort Against Tribulation” (1534) which says, “A drowning man will clutch at straws.” It is said that the “straw” in this case refers to the sort of thin reeds that grow by the side of a river.