as a type of tiny biting insect, 1839, American English, from midge, perhaps with diminutive suffix -et.
Dr. Webster is in error in saying the word "midge" is "not in use" at the present day. In the neighboring Green mountain districts, one or more most annoying species of Simulium that there abound, are daily designated in common conversation as the midges, or, as the name is often corrupted, the midgets. From Dr. Harris' treatise it appears that the same name is in popular use for the same insects in Maine. The term is limited in this country, we believe, exclusively to those minute insects, smaller than the musketoe, which suck the blood of other animals. ["Transactions of the New-York State Agricultural Society," vol. vi, Albany, 1847]
Transferred sense of "very small person" is attested by 1854. It is also noted mid-19c. as a pet form of Margaret.