Seems like it comes to English from the Arabic word for blessed, and has enjoyed many adventures along the way.
OED
[Perh. < an unattested Middle Low German *moneke (inferred from Moneke, the name of a character in Reynard the Fox, attested earlier in a French context as Monnekin, Monnequin (c1330): see note below), prob. a diminutive formation (see -KIN) < Middle French monne (1545; French mone), either < Italian monna, mona (c1547) or its probable etymon Spanish mona (1438; later also mono (1494)), prob. shortened < maymon (1270) < Arabic maym{umac}n (adjective) blessed (13th cent. in Vocabulista, glossed ‘beatus’), (noun) monkey.
The reason for this application of Arabic maym{umac}n is conjectural: it may be a euphemism by antiphrasis (cf. the divine injunction in the Qur’an to transgressors ‘be ye apes, despised and rejected’). The reason for the loss of the first syllable of the Arabic word is also unclear.
It is tempting to see the single attestation of MANCOWE n. as an earlier variant of the present word, although it is difficult to relate formally to the assumed etymon or to the attested forms of the English word, and the resemblance may be no more than coincidental.
The Middle Low German version of Reynard the Fox (1498) has (only once, l. 6161) Moneke as the name of the son of Martin the Ape; early in the 14th cent. the same character is mentioned as Monnekin (also in a textual variant Monnequin) by the Hainaulter Jean de Condé in Li Dis d’Entendement (Scheler) 853 (the passage is also printed by Chabaille as a ‘branche’ of the Roman du Renart). As the name does not occur in any other version of Reynard, the English word is very unlikely to be derived from the story. But the proper name may represent an otherwise unrecorded Middle Low German *moneke, Middle Dutch *monnekijn, a colloquial word for monkey, and this may have been brought to England by showmen from the continent.
Romance diminutive forms include Middle French monin (masculine, c1345), monine (feminine, mid 16th cent.), Old Occitan monina (1470; Occitan monin (masculine), monina (feminine)), Italian monnina (16th cent. in form monnino) and {dag}monicchio (Florio, 1598).
With Spanish forms which exhibit loss of initial syllable, cf. Occitan mouno female ape, Portuguese mona (16th cent.). Spanish mona or Portuguese mona {goesto} the scientific Latin specific name for a type of guenon: see MONA n. Cf. also MONE n.4
With Spanish forms which do not exhibit loss of initial syllable, cf. post-classical Latin maimo (11th cent.), Catalan maymon (1284), Old French maimon, mainmonnet (c1290 as plural mainmonnes; 1351 in Middle French as memon; 15th cent. as mimonet, mimmonet), Old Occitan maimon (1339), Italian mamone, mammone (13th cent.; mid 15th cent. as maimone).
OED