I have come across the  most convincing story yet of the source of the word oyinbo. Of all the previous stories about the source of the word, the current entry in Wikipedia makes use of cognates and probable historical anecdotes. See below:

it… means “white person” generally used in the country, which originates from Igbo Language . The first “White people” to settle in Nigeria as colonial masters were the English people. In Igbo language demonym takes the form ” onye + the place of origin” of the person, hence, and Igbo person is called ” Onye Igbo”. A Yoruba person is called ” Onye Yoruba”. A German is “onye Germany”. Thus the first white people were called “onye ocha” for singular and “ndi ocha” for plural meaning “white person and white people” respectively. This was because the Igbo people of those days did not know from where the white people came. Interaction between the Igbos and the white people resulted in the white people trying to refer to the Igbos with a name similar to what the Igbos called them, but there was problem in pronouncing Igbo words due to presence of double lettered alphabets which involve nasal pronunciation,in some of the consonants such as ‘ch’, ‘gb’, ‘gh’, ‘gw’, ‘kp’, ‘kw’, ‘nw’, ‘ny’, ‘sh’. These were not present in English language hence the difficulty in the white man’s effort in giving the Igbos similar demonym as the Igbo people had given to him, instead a name resulting from a mutilation of Igbo words was produced “Oyi ibo’ instead of ” onyi igbo’ meaning ‘Igbo person’ just as he ‘the white man’ was called ‘ onye ocha’ meaning ‘white person’. It was this ‘oyi ibo’ that the Igbos later started referring to as ‘white person’ in a way of mocking the white man for his inability in saying “Onye Igbo”. This would later be adopted by other Southern Nigerian tribes as the standard name for the white man and coupled with dialect variance one obtains different pronunciations such as “Oyinbo’ in Yoruba and other western Nigerian tribes…

As a linguist, this makes more sense than any other story that breaks the word down as “oyin + bo” or any other permutation.