From 'The Insect that Stole Butter' - Oxford Dictionary of word origins
(everyone should have this entertaining book)
Jim Carroll

SALAD [Late Middle English] One of many words that go back to Latin sal ‘salt. The root implies that it was the dressing or seasoning that originally characterized a salad, and not the vegetables. The expression your salad days, 'the time when you are young and inexperienced’, is one of Shakespeare’s inventions, occurring in Antony and Cleopatra. The idea behind the phrase becomes clearer when you read the full line spoken by Cleopatra: ‘My salad days, When I was green in judgement'. Shakespeare used the word salad in a play on ‘green, which is still used today in the sense 'inexperienced or naive’. The expression was made better known by the success of Julian Slade’s 1956 musical Salad Days about some students starting out in the adult world.