Abstract Introduction Slow wave sleep (SWS) strengthens memory for studied information, but research on the effect of sleep on information that is not intentionally remembered is scarce. Previous research from our lab suggests sleep consolidates some information that has been encoded incidentally, meaning that it has been acted on but not intentionally remembered. It remains unclear what determines which information is consolidated during sleep after incidental encoding and what aspects of sleep are related to this mnemonic benefit. In two experiments, we test the hypothesis that sleep consolidates strong but not weak memory traces following incidental encoding and assess the relationship between memory performance and sleep attributes. Methods In Experiment 1, we manipulated memory strength within- and between-subjects. Participants rated words one or three times (within) in a shallow or deep incidental encoding task (between). In the shallow task, participants counted vowels in each word; in the deep task, participants rated each word on a scale from ‘concrete’ to ‘abstract’. Following a 12-hour period containing sleep or wakefulness, participants took a surprise memory test. In Experiment 2, participants rated words one or three times in the deep encoding task, received an 8-hour sleep opportunity with partial PSG, and took the surprise memory test. Results In Experiment 1, participants remembered words better after sleep than wake regardless of number of encoding trials, but only after deep encoding. There was not an effect of sleep following shallow encoding. In Experiment 2, SWS correlated negatively with response latency for correctly recognized words encoded once, but not those encoded three times. That is, participants who received more SWS showed faster performance. Conclusion Results suggest sleep consolidated information based on the depth of encoding, and this benefit was related to SWS. This work is broadly consistent with theories of memory consolidation that predict sleep is more beneficial for strong than weak memory traces, such as the synaptic downscaling hypothesis. Support (if any):