1. A species of herring of the eastern North Atlantic
2. A small or inconsequential person or thing
Jack Sprat could eat no fat,
his wife could eat no lean
Put them both together
and they lick the platter clean.
This poem goes back to at least the mid-17th century and calling a person of small stature Jack Sprat goes back to the 16th century. Sprat has a few definitions and a couple different etymologies, depending on what it means.
The oldest definition is "an evil spirit" and it dates to the mid-15th century. The Oxford English Dictionary calls its origin 'obscure' and compares it to scrat, which means "a hermaphrodite" and comes from Old English scratta. Scratta derives from Old High German scrato ("satyr, wood-demon") and/or Middle High German schrat ("goblin, elf"). This etymology does not appear to be connected to the definitions given above.
Both the fish and the "small" definitions come from the same source: sprot. Sprat is a variant of sprot that emerged by the mid-16th century. Sprot is an Old English word that presumably comes from some Germanic source because there are cognates in Frisian, Dutch, Danish, and German. Sprot has all the same definitions as sprat plus "a smelt."
As for where the sprat in Jack Sprat comes from, your guess is as good as mine. I'd guess the latter etymology because connection between "small" and "diminutive." But the former etymology does link up with "elf," so it could go either way.
Ever notice how busy 'Jack' is in fairy tales and nursery rhymes? Find out why here.