Cardinal sine function or sinc function is defined by: \begin{equation} \mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases} \end{equation}
Is there any established name and notation for "hyperbolic sinc" function, where $\sin x$ is replaced by $\sinh x$?
Better late than never. I would refer to it as the "hyperbolic sinc."
The notation I've seen used is sinhc x.
There's an article Here, which provides a geometric interpretation for it.
Full Reference:
The Hyperbolic Sine Cardinal and the Catenary, by Javier Sa ́nchez-Reyes, The College Mathematics Journal, Vol. 43, No. 4 (September 2012), pp. 285-290 http://dx.doi.org/10.4169/college.math.j.43.4.285