What component can I use with an Arduino to measure the capacitance of objects?
You can use the Arduino to measure capacitance using an RCtime approach (Google that for code examples), which requires that you have a known resistance for reference.  You should also have some known capacitors to make sure you have everything calibrated properly.
But it sounds like what you really want is more like &quot;dielectric constant&quot;.  You can use
the same setup to measure that, by creating a standard reference capacitor consisting of two conductive plates of known area that you sandwich your unknown dielectric material between.  The value of capacitance is related to the surface area of the plates and to how close together they are... closer gives more capacitance.  Look up the formulas for this.
This is a standard Physics 101 experiment.
The problem is that unless you get very thin sheets of the material, you are not going to get a very high capacitance, which will make it hard to measure.  This is not going to be a trivial project.