The Oxford English Dictionary has no mention of New York's Soho, but has this for London:
The name of a district in the West End of London, noted for its foreign population, prostitutes, and restaurants, and latterly for its night clubs, striptease shows, pornography shops, etc. Freq. attrib. of things connected with or characteristic of Soho.
1818 KEATS Let. 14 Mar. (1931) I. 127 Then who would go Into dark Soho, And chatter with dack'd hair'd critics.
The Oxford English Dictionary defines the term "soho" to this, from evidence that goes back to the 14th century:
A call used by huntsmen to direct the attention of the dogs or of other hunters to a hare which has been discovered or started, or to encourage them in the chase; hence used as a call to draw the attention of any person, announce a discovery, or the like.
New York took the name in the late 1960s.
19 October 1969, New York Times, pg. D29:
SOHO
What's so special about the South Houston Industrial Area (known in planning jargon as SOHO), a 40-block district bounded by Houston St. on the north, Canal on the south, West Broadway on the west, and Lafayette on the east? For one thing, it coincides with one of the city's finest architectural areas, the cast-iron district. And for another, the spacious loft buildings that once harbored mostly small businesses have been infiltrated by thousands of artists and their families.

