We present a comprehensive characterization of the memory effect arising in thin-junction silicon Single-Photon Avalanche Diodes (SPADs) when exposed to strong illumination. This partially unknown afterpulsing-like noise represents the main limiting factor when time-gated acquisitions are exploited to increase the measurement dynamic range of very fast (picosecond scale) and faint (single-photon) optical signals following a strong stray one. We report the dependences of this unwelcome signal-related noise on photon wavelength, detector temperature, and biasing conditions. Our results suggest that this so-called "memory effect" is generated in the deep regions of the detector, well below the depleted region, and its contribution on detector response is visible only when time-gated SPADs are exploited to reject a strong burst of photons.