The story
Apparently coined by a Dr. Tufts of early 18th-century London, as the name of one of five diseases his medicine was claimed to be able to cure. According to C.J.S. Thompson's "The Quacks of Old London" (page 110), the marthambles is one of several nonexistent diseases invented by Dr. Tufts in a pamphlet circa 1700 in order to sell his tonics and medicines. Other diseases mentioned in Tufts' pamphlet are the "Strong Fives", the "Moon Pall", and the "Hockogrockle". Tufts claimed to have encountered these diseases on his travels over a period of forty years, and that he could cure them all.
Etymology from Wiktionary, CC BY-SA 4.0.