So the person primarily, but also what he or she says. First attested around 1650, but there's a bit of disagreement by authorities as to the exact origin -- the blather- part being, indeed, blather (from Middle English, from Old Norse blathra, to chatter/blabber), but some assert the original form of the second half is -skite, from Old English meaning shit (via a Middle English form meaning diarrhea) while others it's -skate (sometimes spelled -skite) meaning contemptible person (as in cheapskate, of uncertain origin). In any case, all agree the nonsense sense was originally an Americanism, arising because of its use in the popular song "Maggie Lauder," especially among the soldiers of the Continental Army during the War of Independence.
"Mr. President, you are a blatherskite!" he said, and stalked out.
(Originally said to Franklin Delano Roosevelt by one Samuel Seabury.)