The name is a corruption of its original name, Convent Garden, which gives a clue as to the site's original purpose. I'd make that clue more obvious, but I'm worried I'd make a "habit" out of it (megalolz). Once a garden belonging to Westminster Abbey, the land was tilted and worked by Benedictine monks, until King Henry VIII seized the land in 1536—a consequence of Henry's separation from the Catholic Church. The land was in turn granted to John Russell, the 1st Earl of Bedford, who showed little interest in the land and left it unattended. Don't tell anyone living in a "compact and bijou" million-pound studio flat that there was a point in time where people didn't give a toss about London real estate. The land stayed in the Russell family, unused, until the 4th Earl of Bedford inherited the land in 1627. He hired Inigo Jones to design and build a piazza on the property—the project, inspired by the piazzas of Italy and the first of its kind in England, was completed in 1631, and remains to this day the very best place in England to go if you want to see men sprayed silver standing very still and expecting to be paid for it.
The earliest reference to Covent Garden as a market dates back to 1654. For roughly 16 years, the market that operated there was an informal organization. Samuel Pepys reports seeing an Italian puppet show in the piazza in 1662, indicating more activities occurred at Covent Garden at this time than just trade. King Charles II formally granted the Earl of Bedford market rights in 1670, officially allowing the landowner to profit from the business that occurred on his property.
The Covent Garden Market was moved to a new location south of the Thames in 1974.