55 Cnc e is an ultra-short period super-Earth transiting a Sun-like star. Previous observations in the optical range detected a time-variable flux modulation that is phased with the planetary orbital period, whose amplitude is too large to be explained by reflected light and thermal emission alone. The goal of the study is to investigate the origin of the variability and timescale of the phase-curve modulation in 55 Cnc e. To this end, we used the CHaracterising ExOPlanet Satellite (CHEOPS), whose exquisite photometric precision provides an opportunity to characterise minute changes in the phase curve from one orbit to the next. CHEOPS observed 29 individual visits of 55 Cnc e between March 2020 and February 2022. Based on these observations, we investigated the different processes that could be at the origin of the observed modulation. In particular, we built a toy model to assess whether a circumstellar torus of dust driven by radiation pressure and gravity might match the observed flux variability timescale. We find that the phase-curve amplitude and peak offset of 55 Cnc e do vary between visits. The sublimation timescales of selected dust species reveal that silicates expected in an Earth-like mantle would not survive long enough to explain the observed phase-curve modulation. We find that silicon carbide, quartz, and graphite are plausible candidates for the circumstellar torus composition because their sublimation timescales are long. The extensive CHEOPS observations confirm that the phase-curve amplitude and offset vary in time.We find that dust could provide the grey opacity source required to match the observations. However, the data at hand do not provide evidence that circumstellar material with a variable grain mass per unit area causes the observed variability. Future observations with the James Webb Space Telescope promise exciting insights into this iconic super-Earth.