The pub was named after John Wilkes, who was held in high esteem for "writing down" the Marquis of Bute.

To be sold, a Freehold Tenement situate in the Fore Hamlet in Ipswich, lately known by the Sign of Wilk's Head.Ipswich Journal, 6th Aug 1774***

To be lett or sold, a Tenement, situated in the Hamlet in Ipswich, formerly known by the Sign of the Wilkes's Head.Ipswich Journal, 25 Feb 1775***

It was probably renamed after this sale was completed (presumably it's then one of the other pubs listed in the area).