This phrase is sometimes misspelled as 'show your metal'. In fact, that's not really so far wrong. Metal and mettle were originally variations of spelling for the hard, shiny substance we now always spell 'metal'. Yes, I know that not all metals are hard, but the scientific definition of what is or isn't is rather laboured.
The first known use of a variant of 'show your mettle' is found in John Fletcher's Monsieur Thomas, 1619:
"When did he ride abroad since he came over? What Tavern has he us'd to? What things done That shews a man, and mettle?"
Until the end of the 17th century the two spellings, 'mettle' and 'metal', were virtually interchangable and sometimes both occurred in the same text, as in Daniel Rogers' Naaman the Syrian, his disease and cure, 1642:
"Then she shewes the metall she is made of."
"To try the spirit of men, of what mettle they are made of."
By the turn of the 18th century though the two spellings had begun to diverge. 'Mettle' was usually reserved for 'character, disposition - the stuff we are made of', for example:
The Free-thinker March 1719: "I like the Lady's Wit and Mettle."
'Metal' was more often used as we use it now, for example:
Encyclopedia Brittanica, 1797: "To free the noble metals from the stony matter..."
Of course, there were exceptions and poets looking for a rhyme with kettle could hardly be expected not to be tempted by mettle.