Any material used in diaphragm will have properties that require attention to gain optimal operation. Beryllium as an example has very high self damping (stiffness of cone) but doesn't exhibit much internal damping (ability to dissipate excess energy within the material). While the operating range of one will be greater, care to ensure you stay within its operating range for its given size and construction.
I would like to address carbon fiber, since its brought a number of times but aspects of it should be known. There are so many grades of carbon fiber and the epoxies used, construction method employed, and design parameters in that you can come to some very wildly different results with the very same material. You could spend an considerable amount of time in FEA design between differing grades, weaves, shapes, etc. The modulus alone could vary by a factor of five just by simply changing from a low modulus to high modulus fiber. The expense of high modulus fiber is notably more and typically forgoes some tensile strength when modulus increases. But in this application, tensile strength matters little with it being well beyond what is required in nearly all grades. The latest development in this area has been the infusion of graphene in the epoxy, which aids in producing a more uniform carbon fiber sheet. This will further extend how light a driver cone could be made without any loss of stiffness. This still doesn't account for the mass of the voice coil, former, surround, etc.
So when your looking for the best material, it must include a completed design. Only with this, combined with the desired operating range and crossover implementation could one select the best material within a given cost. I do mention cost since its very easy to gain a marginal improvement for an extreme cost, one which is better spent in other areas of the system.
