Up in arms
The original usage of 'up in arms' was entirely literal. To be 'in arms' or 'at arms' was to be equipped with weapons and armour. It isn't clear why 'arms' was chosen as the name for weaponry. It may be as simple as a sword or club being seen as an extension of the arm. 'Armour' is just a form of defensive weaponry that a soldier was clad in. Like 'vesture', meaning 'that which a person is dressed in', that is, clothes, the 'ure' part may be translated as something like 'collection of'. The spelling would be more properly 'armure', which is how it was spelled in early texts; for example: Robert of Gloucester's Metrical Chronicle, 1297:
He & hys armure...
The style and decoration of armour was how knights were distinguished from one another in battle. This was important, as knights were more often captured and later ransomed rather than killed. Heraldic arms were the formalised development of that identity. A family's 'Coat of Arms' was originally a actual decorated coat, worn by a knight when in battle. In the romantic poem Sir Gawain and the Green Knight, circa 1340, Gawain is described as being dressed "With ryche cote armure".
There are many phrases in English that refer to 'arms' - 'man at arms', 'firearms', 'to arms!', bear arms', and 'up in arms' adds to that list. It is merely the more active form of 'in arms', in that it indicates that a soldier wasn't just equipped for a fight, he was on his feet and ready for it.
The term 'up in arms' began to appear in print in the 1590s. Who coined it isn't entirely clear, although it does sound Shakespearian and the Bard did use it in more than one play of the early 1590s - Henry VI Part II:
The princely Warwick, and the Nevils all,
Whose dreadful swords were never drawn in vain,
As hating thee, are rising up in arms:
And now the house of York, thrust from the crown
By shameful murder of a guiltless king
And lofty proud encroaching tyranny,
Burns with revenging fire; whose hopeful colours
Advance our half-faced sun, striving to shine,
Under the which is writ 'Invitis nubibus.'
The commons here in Kent are up in arms:
and Richard III:
March on, march on, since we are up in arms;
Another contender is Sir Thomas More, who used the term in a work dated circa 1590:
A number poore artificers are up In arms.
The date of writing of More's piece and of both of Shakespeare's plays in debatable, so the author can't be definitively decided.
See other phrases and sayings from Shakespeare.