Writer Matthew Stewart responds to Tuck Dean Paul Danos' letter in this month's Atlantic by stating that "the Tuck School at Dartmouth was indeed founded first, as Paul Danos says; but Harvard was first to offer a master's degree in business administration."
This is a petty point, but it's not clear how Stewart could be correct if he's referring to the degree in lower-case letters: the Amos Tuck School of Administration and Finance was founded in 1900 and granted its first business master's degree to Walter Blair and his two classmates in 1901. Harvard did not admit its first students until the fall of 1908.
No graduate degree in the administration of modern businesses existed in 1900. The Tuck School started by calling its degree the "Master of Science (Tuck School)" but renamed it during 1902 the "Master of Commercial Science," a reference to the Bachelor of Commerce degree that many schools had been awarding for years. (Wayne Broehl, Tuck & Tucker (1999), 43-44.)
Harvard's business school (which also might have begun by calling its degree the M.C.S.) did not seem to rename whatever graduate degree it was granting as a "Master of Business Administration" until 1921, and that's the name that caught on. (Harvard probably did more than simply rename its existing degree -- it probably invented a wholly-new curriculum that revolutionized business education; but Harvard did not open "the first graduate school in the country to offer a master's degree in business" as Stewart stated in the original article last month.) In 1953, the Tuck School changed the name of the degree it had been awarding since 1901 to the conventional M.B.A.
[09.12.2006 update: reworded post, fixed typo.]