It was due to the development directed by The Irish Society towards rebuilding and expanding the city, that it was renamed Londonderry in honour of the capital and colonisation from London.
A pageant entitled "The Triumphs of Love and Antiquity" was performed for Cockayne's mayoral inauguration on 29 October 1619, written by Thomas Middleton.
William Baffin was equipped for one of his northern voyages by Cockayne and others of the Merchant Adventurers' Company and a harbour in Greenland was named in his honour, called 'Cockin's Sound' on the Admiralty chart.
He died on 20 October 1626, in his sixty-sixth year, at his manor house at Comb Nevill in Kingston, Surrey, and was buried in Old St Paul's Cathedral,[3] where his funeral sermon was preached by John Donne and a monument was raised to him.
He married Mary Morris on 22 June 1596 in London, and they had seven children together: His widow remarried, 6 July 1630, Henry Carey, 4th Baron Hunsdon, 1st Earl of Dover, a great-great grandson of Thomas Boleyn, 1st Earl of Wiltshire, father of Anne Boleyn and, dying 24 December 1648, was buried with her first husband at St. Paul's.