Human activity recognition (HAR) is a classification problem involving time-dependent signals produced by body monitoring, and its application domain covers all the aspects of human life, from healthcare to sport, from safety to smart environments. As such, it is naturally well suited for on-edge deployment of personalized point-of-care (POC) analyses or other tailored services for the user. However, typical smart and wearable devices suffer from relevant limitations regarding energy consumption, and this significantly hinders the possibility for successful employment of edge computing for tasks like HAR. In this paper, we investigate how this problem can be mitigated by adopting a neuromorphic approach. By comparing optimized classifiers based on traditional deep neural network (DNN) architectures as well as on recent alternatives like the Legendre Memory Unit (LMU), we show how spiking neural networks (SNNs) can effectively deal with the temporal signals typical of HAR providing high performances at a low energy cost. By carrying out an application-oriented hyperparameter optimization, we also propose a methodology flexible to be extended to different domains, to enlarge the field of neuro-inspired classifier suitable for on-edge artificial intelligence of things (AIoT) applications.