目 (eye) anchors the meaning: sleep is what eyes do when closed. Placing 眠 inside the eye-radical family ties it to 睡 (sleep), 眨 (blink), 眯 (squint), and 瞎 (blind) - all states of vision.
民 (mín) supplies the sound, drifted to mián. There's a quiet historical irony in the choice - early forms of 民 depict a pierced eye marking a captive, which lurks behind the eye-radical here. Same phonetic series: 眠, 泯, 抿.
