While a mystery is תעלומה, a detective is בלש, so that we have:
הבלש עלה על משהו שישפוך אור על התעלומה.
The detective caught onto something that will shed light on the mystery.
If the detective is a woman, she’s a בלשית.
בלש and בלשית come from the Aramaic root ב.ל.ש meaning “searching”, which is what detectives spend their time doing.
A related word is בלשנות – linguistics – a field called such after a biblical name מרדכי הבלשן (Mordechai the “balshan”), whome Jewish lore describes as someone who knew many languages.
A private investigator, also a type of detective, is חוקר פרטי.
