verification, effect, testing
驗 centers on checking something to confirm it is true or works: verifying, testing, or examining for an effect. The idea of proof or evidence (a sign, a mark) extends naturally from that act of verification.
驗 combines 馬 (horse) with 僉, which likely contributes the sound. The original idea may relate to inspecting or testing horses, though the exact historical development is uncertain.
A horse (馬) is being examined by a person with a checklist (僉, which looks like a gathering of people under a roof). Picture a vet verifying the horse's health: testing, checking, and looking for signs.
For ケン, imagine Ken the inspector verifying the horse: Ken -> ケン, and he tests everything carefully.