Sean, the other guys have given some useful information. Here is a link to a thread which gives some nice pictures and discusses different materials others have suggested from plywood to FR4/G10.
http://www.thevirtualboatyard.com/2009/11/installing-backing-plates.html
The FR4/G10 glass epoxy is actually pretty similar to the basic materials used to make a fiberglass boat although obviously the reinforcing mat is different.
Others have suggested steel and aluminum. All these materials can be used but as noted in the post there can be some challenges with machining/cutting or corrosion due to different materials etc. All these materials are available from e-bay to save the time rummaging around Home Depot or ? to find the size you need.
There has been some discussion about the effectiveness of say HDPE, UMHM and say Starboard. FYI here are a couple of data points
The compressive strength of the materials in kpsi (Thousand of psi) are
Material Compressive strength Modulus
Stainless steel (type 304) 31 ksi
Aluminum (type 6061-T6) 40 ksi
FR4/G10 65 ksi
Delrin 16 ksi
Starboard (HDPE) 4.6 ksi
From this data you can see that the FR4/G10 material is actually stronger than the others. The strengths given for the metals is the yield strength.
Since are also looking for stiffness in a backing plate we might also compare the modulus of elasticity; so for the same list here is the modulus. For the polymers I have provided the compressive modulus since it is different that either the tensile or flexural.
Stainless steel (type 304) 30 million psi
Aluminum (type 6061-T6) 10 million psi
FR4/G10 about 2.6 million psi
Delrin 450,000 psi
Starboard (HDPE) 100,000 psi
Anyway I hope this information help you make a decision about the solution you choose