Once upon a time, you could have told me Jack Robinson was a jack in the box, the name meant nothing to me.

Not until Phryne Fisher came along, a rather brilliant 1920s private detective series set in the back streets of Melbourne, as well as more salubrious houses of the rich and famous.
In this series, there is a policeman, a foil for her detective moments, and a love interest that is always just beyond her grasp, a man by the name of Inspector Jack Robinson.

How coincidental.

But…

As for the saying, before you can say Jack Robinson…

It has nothing to do with Phryne Fishers Inspector.

Instead,

There is one story of a politician, Jack Robinson, in the late eighteenth century who was accused of bribery on the floor of the house of commons in England. His accuser was another MP who was asked to name the culprit, and thereby coined the term, ‘I could name him as soon as I could say Jack Robinson’.

The second was a Jack Robinson, the hero of a story written in the nineteenth century who came home to find his intended wife married to another, and to assuage the pain of it was back to the sea, ‘afore you could say Jack Robinson’.

I’m sure there’s a ton of other saying that could be attached to the name, but these seem to be the accepted reason for the term ‘before you can say Jack Robinson’.