By virtue of recent developments in brain measurement technology, it is now recognized that information processing in brain includes not only neurons but also astrocytes. For this reason, to illustrate the effects of astrocyte on information processing in neuronal systems, we research the weak signal detection performance of the Hodgkin-Huxley neuron under the effect of astrocyte. It is found that the weak signal detection performance of the neuron exhibits the stochastic resonance phenomenon depending on noise intensity, where the presence of astrocyte with an optimal coupling strength significantly increases the detection performance of the neuron when compared the one without astrocyte. The obtained results also reveal that there is an optimal weak signal frequency ensuring the best detection performance. Besides, we show that the colored noise exhibits a better performance than white Gaussian noise on improving the weak signal detection capacity of the neuron; moreover, the weak signal detection performance of the neuron demonstrates a resonance-like dependence on the correlation time of the noise. Finally, we investigate the effects of calcium channel noise. Although the calcium channel noise generally reduces the weak signal detection performance of the neuron, the optimal coupling strength warranting the best detection performance critically depends on its intensity.