The spelling of "be thinks" is actually incorrect. The correct phrase is "methinks", which is an old-fashioned way of saying "I think". It comes from the Old English phrase "me thinks", which translates to "it seems to me". The "me" is simply the object pronoun, while "thinks" is the third person singular form of the verb "think". In IPA phonetic transcription, "methinks" is pronounced /miːˈθɪŋks/.
The phrase "be thinks" does not have a separate etymology, as it is a grammatical construction that combines the verb "be" with the Old English reflexive pronoun "thinks".
In Old English, the verb "thyncan" (meaning "to seem" or "to think") could be conjugated with the reflexive pronoun "me" (meaning "myself"), resulting in "mē thinks" which literally translates to "it seems to me" or "I think". This construction was commonly used to express personal opinions or impressions.
Over time, the reflexive pronoun "me" merged with the verb "thinks" and eventually evolved into the form "methinks". However, in modern English, this construction is rarely used, and only the pronoun "me" has survived.