Call an n-gon "rational" if all edge lengths are rational.
Call an n-gon "rational-unit-cyclic" if it's rational and can be inscribed in a unit circle.
Call an n-gon "primitive-rational-unit cyclic" (PRUC) if it's rational-unit-cyclic and, for any reordering of the edges, no diagonal has rational length.
Note: In my previously posted tentative definition, there were additional requirements, namely: n > 3, pairwise distinct edge lengths, and no edge length equal to 2. Those additional requirements have now been dropped.
Any rational-unit-cyclic triangle is PRUC, and the class of all such triangles can be represented parametrically.
There exist PRUC quadrilaterals. For example, the quadrilateral found by Richard Tobin with sides 8,36,57,62 and radius 32 is a PRUC quadrilateral scaled by a factor of 32.
There are other PRUC quadrilaterals as well, but there's no clear pattern that I can see relating the numerical values of the edge lengths. It's not clear how to generate them except by brute force search.
So that's the question.
For some n > 3, either general or specific, is there some subclass of the class of PRUC n-gons which can be generated by a method other than brute force search? Perhaps a parametric representation or a recursion?
In the meantime, the current search methods might help by revealing a pattern for some subclass of PRUC n-gons.