Although puisne, meaning "junior" or "inferior" is known in legal English from the early Middle Ages, the spelling puny didn't appear until Shakespeare's time, when it developed two distinct senses. When Richard II refers to Bolingbroke as a "puny subject" (Richard II, III.2.82), or Othello says "Every puny whipster gets my sword" (Othello, V.ii.244), this is the modern sense: Othello means "pathetic wretch". But when the Bastard of France reminds everyone of how Young Talbot "Did flesh his puny sword in Frenchmen's blood" (Henry VI Part I, IV.vii.36) it can hardly mean "feeble", if the sword is doing such a great job. Rather it is a development of the "junior" sense - "untried, raw, inexperienced". This was the first time Young Talbot had been in battle. The two senses nicely interact when Prince Hal describes Francis the tavern-waiter as a "puny drawer" (Henry IV Part I, II.iv.29).
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin