目 depicts a vertical eye with pupil - originally rotated from a horizontal eye in oracle bone script. As radical it marks 盹 as something the eyes do. Joins 看睡眠瞌 - all sight or sleep characters.
屯 supplies the sound (tún drifted to dǔn). The same phonetic appears in 钝顿囤. A faint suggestion of stillness and gathering fits the image of eyes settling momentarily - a brief nap rather than full sleep.
