BOOST (BOOst Symmetry Test) is a proposed space mission to search for Lorentz invariance violations and aims to improve the Kennedy-Thorndike parameter constraint by two orders of magnitude. The mission consists of comparing two optical frequency references of different nature, an optical cavity and a hyperfine transition in molecular iodine, in a low Earth orbit. Naturally, the stability of the frequency references at the orbit period of 5400 s (f=0.18 mHz) is essential for the mission success. Here we present our experimental efforts to achieve the required fractional frequency stability of 7.4×10-14 Hz -1/2 at 0.18 mHz (in units of the square root of the power spectral density), using a high-finesse optical cavity. We have demonstrated a frequency stability of (9±3)×10-14 Hz -1/2 at 0.18 mHz, which corresponds to an Allan deviation of 10-14 at 5400 s. A thorough noise source breakdown is presented, which allows us to identify the critical aspects to consider for a future space-qualified optical cavity for BOOST. The major noise contributor at sub-milli-Hertz frequency was related to intensity fluctuations, followed by thermal noise and beam pointing. Other noise sources had a negligible effect on the frequency stability, including temperature fluctuations, which were strongly attenuated by a five-layer thermal shield.