For the non-medical crowd, Gilbert's syndrome is a benign, familial condition in which the liver does not fully process a breakdown product of hemoglobin, bilirubin.

We can't be sure that Napoleon had Gilbert's syndrome, since to meet modern diagnostic criteria, we would need laboratory evidence of unconjugated hyperbilirubinemia (high levels of the form of bilirubin that has not been processed by the liver) and the absence of other likely causes of that hyperbilirubinemia. Failing that, we would need to sequence his DNA for the mutation that causes Gilbert's. Neither option sounds feasible.

We probably say Napoleon has Gilbert's syndrome because...Gilbert said so. In the paper where he first characterizes the syndrome that would come to bear his name, he makes the case for Napoleon having the syndrome. Gilbert notes that Napoleon's mother had gallstones, which he implies is related to the syndrome and thus evidence of a family history, and that Napoleon himself was often jaudiced and itchy--both of which we still accept today as indicators of hyperbilirubinemia. Gilbert also claims that Napoleon had other signs, like sleepiness and bradycardia (slow heart rate), which he believed were linked to the syndrome (we don't think that's the case now).

Like any good anecdote, this has probably been carried forward through the years based on Gilbert's assertion.

Aster, Jon, Vinay Kumar, Abul K. Abbas, and Nelson Fausto. "Robbins & cotran pathologic basis of disease." Philadelphia, Saunders 54 (2009).

Gilbert, Augustin and Lereboullet, P. "La cholemie simple familiale." Semin Med 21 (1901): 241-243.