Is there a name for a hyperbolic quadrilateral where opposite sides have equal length, or equivalently so that the diagonals bisect each other, as shown below?
"Hyperbolic parallelogram" is the obvious name (and it seems to be used a little), but has the obvious downside that these quadrilaterals generalize many properties of Euclidean parallelograms, but not the property that gives the "parallelogram" its name.
In a paper I need the easy lemma that $$ \cosh (OX)\cosh(OY) = \frac{\cosh(XY) + \cosh(YZ)}{2} $$ generalizing the Euclidean parallelogram law. In general these quadrilaterals seem pretty pleasant to work with.
If you're looking for a self-describing term, consider point-symmetric quadrilateral. Alternatively, rhomboid is an uncommon term for parallelograms, but traditionally only refers to parallelograms that are neither rectangles or rhombi.