I have always pronounced it 'shine'. I really had no formal exposure to it, instead learning about hyperbolic functions on my own. cosh=coshine made sense to me, and thus sinh=cosh-co=coshine-co=shine. 'cinch' has seemed contrived because of the extra 'c' sound.
By the way, I normally pronounce tanh 'thangent' for the same reason. On the other hand, if I had to pronounce sech, I'd say 'sechant' because 'ch' makes sense as a sound, unlike 'nh'.