There are many products that can be used for noise attenuation, and the objective for your application should be to find the lightest product that gives the greatest attenuation at the frequencies your aircraft vibrates at. It is very, very easy to add lots of weight while producing only tiny improvements in the noise level, so if you choose to pursue this it will be worth your time to do a thorough investigation.
In an earlier life I spent a fair amount of time dealing with resonance issues on small steel structures. For the problem I was dealing with, the best solution turned out to be a 3M viscoelastic polymer bonded to a thin substrate (in this case it was a stainless steel foil). The combination of the polymer and the steel was tremendously effective in damping out the resonant frequencies, and the material worked over a wide range of temperatures. While working on the solution, I found that 3M was selling a similar product to Boeing. Boeing was applying sheets of the material to the inside surface of the fuselage. This product was optimized to damp out the resonant frequencies experienced in a jet, so it might be suboptimal in a Cub, but there's no harm in checking with the Boeing surplus store to see if they have any on hand. The 3M website may also be able to refer you to converters who sell their viscoelastic material bonded to various substrates.
Beech has extensive noise reduction programs for its aircraft. They use passive and active noise damping techniques. Are there any Beech employees on the site who can share the techniques used on Bonanzas and Barons?
The low cost automotive approach is a fibrous mat coated in a tar-like substance. I know this stuff is available through body shops, but I have never seen any performance data, so I can't tell you how much noise attenuation to expect, or the frequencies at which it's effective. I do know that it's rather heavy. Are there any Big 3, BMW, or Mercedes employees on the site who know more about those products?
Eric