We propose to couple a single-pixel camera with a photon counting board in order to obtain an inexpensive time-resolved imaging system having a high spatial and temporal resolution. As an alternative to compressive sensing, a wavelet-based adaptive acquisition strategy is employed which allows high compression rates with little degradation of the image quality. The applicability of our approach is demonstrated for fluorescence lifetime imaging. Experimental results obtained by imaging samples embedding several fluorophores are provided. The proposed imaging system with the wavelet-based strategy can be suitable for a microscope in order to perform fluorescence lifetime imaging microscopy measurements.