‘Brewer’s Dictionary of Phrase and Fable’ offers two or three possibilities. He might have been ‘a very volatile gentleman of that name who used to pay flying visits to his neighbours’.
Again, Jack Robinson was a government minister in the late eighteenth century. The playwright Richard Sheridan was also a Member of Parliament, and in 1780 was attacking government bribery. To cries of ‘Name, Name!’ he replied, looking directly at the minister, ‘Yes, I could name him as soon as I could say Jack Robinson.’
A third explanation is that there was a song popular in the early nineteenth century called ‘Jack Robinson’. It tells how the sailor Jack Robinson returned to find his lady married to another:
. . . says she, ‘I couldn’t wait,
For no tidings could I gain of you Jack Robinson.’
‘But to fret and stew about it’s all in vain,
I’ll get a ship and go to Holland, France or Spain.
No matter where; to Portsmouth I’ll ne’er come again,’
And he was off afore you could say Jack Robinson.
Take your pick. They’re all equally likely or unlikely.