We study memory effects as information backflow for an accelerating two-level detector weakly interacting with a scalar field in the Minkowski vacuum. This is the framework of the well-known Unruh effect: the detector behaves as if it were in a thermal bath with a temperature proportional to its acceleration. Here we show that if we relax the usual assumption of an eternally uniformly accelerating system, and we instead consider the more realistic case in which a finite-size detector starts accelerating at a certain time, information backflow may appear in the dynamics. Our results demonstrate the existence of a connection between the trajectory of the detector in Minkowski space and the behavior of information flow. This allows us to inspect the Unruh effect under a new light, making use of the latest developments in quantum information theory and open quantum systems.