pupil
眹 centers on the pupil of the eye, the dark central opening that lets light in.
眹 combines 目 (eye) with 关, which may have contributed sound. The exact historical development is uncertain.
The left side is an eye 目, and the right side 关 looks like a gate or barrier. Imagine the pupil as the gate through which light enters the eye.
For チン, picture a chin resting on a hand while staring intently, the pupil focused: chin -> チン.