colloq. (orig. U.S.).
[< MONKEY n. + BUSINESS n., probably after Bengali bā̃drāmi. Compare modern Sanskrit vānara-karman (< vānara monkey + karman action, work, employment), Hindi vānara-karma.]
বাঁদর bāndara is "monkey" and বাঁদরামি bāndarāmi is "mischievousness; a monkey-trick; monkeyism". (Monkeyism?)
There are some interesting things about this OED entry, as pointed out on Wordorigins. It's given as "orig. U.S.", but the first citation is British:
1858 T. P. THOMPSON Audi Alteram Partem I. lxv. 249 The Native Indian term for the supreme of folly, is ‘monkey business’.
This quote is from Audi alteram partem, letters of a representative to his constituents by Thomas Perronet Thompson, a British MP. As Dave Wilton says, it's possible that the "orig. U.S." is a mistaken holdover from an older version of the entry, before they added the Thompson quote.