Wordorigins.org

The phrase before you can say Jack Robinson means very quickly, in no time at all. But who was Jack Robinson and how did his name become associated with speed? The answer is we don’t really know, but there are many different explanations.

We do know that the phrase dates to at least 1778 when it appears in Fanny Burney’s Evelina:

I’d do it as soon as say Jack Robinson.