I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native English speaker so I don't know, but in my country we call this function "sintsh" with a $sh$ as in "sharp" and a $s$ as in "saw". This seems to be the natural thing to do if you have not a clue how to call this function. Despite, in my country people try to avoid giving this function a name because they are afraid of making a fool out of themselves.
My question is:
What is the standard name for this function and where does this "shine" come from?
Since this question is about pronunciation of $\sinh$, or the hyperbolic sine, I thought it might be useful to see what the authoritative Oxford English Dictionary had to say:
This reveals that there are actually three pronunciations often encountered. From left to right:
I personally was unaware of the third pronunciation, but it seems your question regards the first pronunciation while the one mentioned by Brian M. Scott (and the one I've heard most often myself...maybe it has to do with being in the US or something) has to do with the second one.