Industry standard measure for coaxial cable stiffness?
There are in fact some very low stiffness cables available. You can hold them in your hand and the protruding part of the cable will hang loosely over your hand.

They are typically very expensive, but provide high phase stability, and are often used in test labs (for example when using a network analyzer).

Unfortunately I don't know of any standardized way of spec'ing these cables. The best way to choose them might be to get a salesperson to bring you a sample to look at.

One example is the Megaphase VN series, but other high-end cable manufacturers have similar products.