http://www.businessballs.com/clichesorigins.htm
"niche - segment or small area, usually meaning suitable for business specialisation - the use of the word 'niche' was popularised by the 19th century expression 'a niche in the temple of fame' which referred to the Pantheon, originally a church in Paris (not the Pantheon in Rome). It was built 1754-80 and converted in 1791 to hold the remains of famous Frenchmen; a 'niche' was a small alcove containing a monument to a person's name and deeds. The French word 'nicher' means 'to make a nest'."
The word "niche" is originally French. And "neesh" is how it's prounounced by French speakers, consequently making it the correct way.