Let me just throw a date out there....
Say...1686?
Interesting tho..I always understood Sir William to have spelled his name "Phips"..vs "Phipps".
I'm no hand writing expert..but seems VERY similar to his penmanship.
Being that this is a Fifth Edition, 1774, it couldn't possibly be William's signature. But....
After some research, I have reason to believe it may have been inscribed to Sir William Phips by his adopted grandson, in Memoriam to his discovery of the Nuestra Señora de la Concepción.
Sir William had no children of his own, but adopted the son of his wife's sister, Spencer Bennett. Spencer took the name Phipps as his own and later became Lieutenant Governor of Massachusetts.
Spencer Phipps' only surviving son was Colonel David Phips. He became high sheriff of Middlesex, Cambridge, Massachusetts, and resided there until the Revolution in 1775 forced him to escape to England, after which his home and all of his personal property left behind were confiscated.
It's my belief, after researching in depth, that this set was possibly part of that estate.
I found a letter from David Phips, with very similar handwriting, in a collection at Harvard University. Notice the similarity of the " P "s I have circled in red....
Letter from David Phips to Colonel Jonathan Snelling regarding escort of Governor Hutchinson to Harvard Commencement, 1773 July 12 · Colonial North American Project at Harvard
It would stand to reason, since his father, Spencer Phipps, spelled his name with two P's...that he would inscribe his grandfather's name the same way.