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Recently, some fast radio burst (FRB) repeaters were reported to exhibit complex, diverse variations of Faraday rotation measures (RMs), which implies that they are surrounded by an inhomogeneous, dynamically evolving, magnetized environment. We systematically investigate some possible astrophysical processes that may cause RM variations of an FRB repeater. The processes include (1) a supernova remnant (SNR) with a fluctuating medium; (2) a binary system with stellar winds from a massive/giant star companion or stellar flares from a low-mass star companion; (3) a pair plasma medium from a neutron star (including pulsar winds, pulsar wind nebulae, and magnetar flares); (4) outflows from a massive black hole. For the SNR scenario, a large relative RM variation within a few years requires that the SNR is young with a thin and local anisotropic shell, or the size of dense gas clouds in interstellar/circumstellar medium around the SNR is extremely small. If the RM variation is caused by the companion medium in a binary system, it is more likely from the stellar winds of a massive/giant star companion. The RM variation contributed by stellar flares from a low-mass star is disfavored, because this scenario predicts an extremely large relative RM variation during a short period of time. The scenarios invoking a pair plasma from a neutron star can be ruled out due to their extremely low RM contributions. Outflows from a massive black hole could provide a large RM variation if the FRB source is in the vicinity of the black hole. [ABSTRACT FROM AUTHOR]