The National Synchrotron Light Source II currently under construction at the Brookhaven National Laboratory is expected to provide unprecedented orbit stability in the storage ring in order to fully utilize the very small emittance of the electron beam. The desire to measure the position of such small beams to high resolution imposes stringent requirements on the thermal and structural stability of the supports for the beam postion monitor (BPM) pick-up electrodes located on multi-pole vacuum chambers and more so on those located upstream and downstream of insertion device sources where the beam size is the smallest. Even with tunnel air temperature expected to be controlled to ±0.1°C, low coefficient of thermal expansion materials is required to meet this level of thermal stability. Here, we present the application of these materials to the design of stable supports for radio frequency (RF)-BPMs as well as the methods of testing their performance.