Previous works only focus on the optimization design for the dual-hop cooperative ambient backscatter communication (AmBC) system with single-relay selection. The impact of relay selection on the outage performance of dual-hop cooperative AmBC systems is still missing. Motivated by this, in this paper, we investigate the outage performance of a dual-hop cooperative AmBC system with single-relay selection, where the backscatter link shares the receiver with the cellular link and the harmful direct-link interference (DLI) is mitigated by using successive interference cancellation (SIC). In the system considered, the selected relay has dual functions. One is to forward message for the cellular link, and the other is to act as the radio-frequency (RF) source for the backscatter device (BD). Specifically, after proposing two novel single-relay selection schemes (RSSs), namely reactive RSS and proactive RSS, we derive the closed-form outage probability (OP) expressions for both RSSs, which can be performed in a distributed manner. To gain more insights, the asymptotic OPs at high signal-to-noise ratio (SNR) are explored and the outage performance comparison between the reactive RSS and proactive RSS are also provided. Results show that the proposed reactive RSS is outage-optimal among all possible single-relay selection schemes. The theoretical analysis is validated by Monte Carlo simulations. The results also show that the relay selection scheme, the number of relays, the location of BD, and the reflection coefficient of BD have great impact on the outage performance of cooperative AmBC systems.