Oxforddictionaries offers this definition:

Having or showing dogged determination not to change one’s attitude or position on something, especially in spite of good arguments or reasons to do so

Several sites have the definition, but not even our reliable Phrase Finder posts an origin.

Word Detective has a related phrase that I’d never heard: “took the studs,” meaning to become stubborn and usually said of a mule. WD found a citation in the Dictionary of American Regional English for “take the studs” (or “get” or “have”) comes from 1797, “and it’s clear that it’s a term primarily applied to balky horses and mules, and, by figurative extension, to uncooperative people.” The word “stud” has an obsolete definition “first appearing in Chaucer in the late 14th century, is ‘defiant of destructive agencies or force; strong, stout.'”

listserv.linguistlist.org has a handful of citations:

• 1771 T. SMOLLETT Humphry Clinker II. 169 The captain..becomes stubborn as a mule, and unmanageable as an elephant unbroke. ·
• 1812 M. EDGEWORTH Absentee xiii, in Tales Fashionable Life VI. 260 She was as obstinate as a mule on that point. ·
• 1853 J. Y. AKERMAN Wiltshire Tales 138 As cam and as obstinate as a mule. ·
• 1922 J. JOYCE Ulysses 411 The likes of her! Stag that one is. Stubborn as a mule! ·
• 1923 Nation (N.Y.) 17 Oct. 432 Then there is the Missouri mule. He it was who won the war. 1972 Listener 21 Dec. 858/2 Not for nothing did the idiom ‘as stubborn as a Missouri mule’ come into the language.

So mules have been identified with stubbornness for a long time. Rinker Buck, in his best-selling book The Oregon Trail, writes that mules are smarter than horses and sometimes smarter than humans who try to drive them into danger. “The common phrase ‘stubborn as a mule,’ [is] a classic example of a man ascribing stupidity to the beast instead of to himself.”

According to Phrase Finder, this is one of many derogatory phrases that became popular in the 1980s to indicate someone “having a bit missing,” or “not all there,” as would be a person playing cards without the full deck.

I’ve read a theory that the expression came from a time when a tax charged was charged on the ace of spades and that people chose to play with 51 cards with that card missing to avoid the tax. Wikipedia has a slightly different story. Charles I of England extended stamp duty to playing cards in 1711. One of the cards in the pack, and usually the ace of spades, was marked with a hand stamp. Another story was that some decks of cards were made without aces because the members of the royal court didn’t want a card that outranked the king and queen.

Wiktionary explains the origin was from the days of flintlock and caplock firearms, “…where the half-cock position of the hammer was both a rudimentary safety and the proper position for priming the pan or inserting a percussion cap. The phrase was originally rendered ‘to go off at half-cock’.” The expression obviously came from the occasion time that the weapon unexpectedly fired when it was at half cock. The expression has evolved into describing someone who takes a premature or ill-advised action.

I’m using this expression as a companion to a commentary about saving and interest rates. According to the Phrase Finder its obvious meaning is, “It is useful to save money that you have as it is to earn more. Forms have been around a long time. George Herbert’s Outlandish Proverbs observed in 1633 “A penny spar’d is twice got.” Deciding to save instead of spend puts you a penny up instead of a penny down, thus “twice got.” The expression evolved to the current form, which is said is incorrectly attributed to Benjamin Franklin. The obvious wonderful message of the expression is that saving instead of spending is a really good idea.