While many automatically think of picture books for kids, the 5-minute bedtime story slot is actually where I get my own reading done, and what works best for me are modern literary flash fiction pieces. I keep a collection by my nightstand.
Stories like those in Lydia Davis's 'Can't and Won'tt' are perfect. They're dense little puzzles that give my brain just enough to chew on before sleep, but they're so short they don't risk keeping me up for 'one more chapter.' The emotional tone matters, too. I avoid anything with high tension. A quiet, observational piece about a mundane detail, like Davis's 'The Cows,' is far better than a plot-driven tale.
For this specific need, I've found curated online journals that specialize in microfiction far more useful than traditional short story anthologies. The commitment is so low that even when exhausted, I can finish one, giving that satisfying sense of completion without the sleep deprivation.