In real-world streaming recommender systems, user preferences often dynamically change over time (e.g., a user may have different preferences during weekdays and weekends). Existing bandit-based streaming recommendation models only consider time as a timestamp, without explicitly modeling the relationship between time variables and time-varying user preferences. This leads to recommendation models that cannot quickly adapt to dynamic scenarios. To address this issue, we propose a contextual bandit approach using hypernetwork, called HyperBandit, which takes time features as input and dynamically adjusts the recommendation model for time-varying user preferences. Specifically, HyperBandit maintains a neural network capable of generating the parameters for estimating time-varying rewards, taking into account the correlation between time features and user preferences. Using the estimated time-varying rewards, a bandit policy is employed to make online recommendations by learning the latent item contexts. To meet the real-time requirements in streaming recommendation scenarios, we have verified the existence of a low-rank structure in the parameter matrix and utilize low-rank factorization for efficient training. Theoretically, we demonstrate a sublinear regret upper bound against the best policy. Extensive experiments on real-world datasets show that the proposed HyperBandit consistently outperforms the state-of-the-art baselines in terms of accumulated rewards.