jaw
腮 centers on the lower part of the face: the jaw. It can also refer to the gills of a fish, which are structurally similar to a jaw.
腮 combines the flesh radical ⺼ with 思, which likely provides the sound. The character is used for the jaw or gills, but the exact historical development of this specific combination is uncertain.
The flesh radical ⺼ on the left suggests a body part, and 思 on the right looks like a thinking face with a prominent jaw. Picture a person deep in thought, resting their chin on their hand, highlighting the jaw.
For サイ, imagine a sigh escaping from your jaw when you are tired: sigh -> サイ, and your jaw drops slightly as you exhale.