I was wondering about that: the New Oxford American Dictionary says to muse comes from the French muser, which comes from the Latin musum. The Muse comes form the Latin musa, which comes from the Greek mousa. Those sound close, but my Latin is too weak (and my Greek inexistent) to tell if they were related back in the antique days.
Learn English – Do the verb “muse” and the noun “Muse” have a common etymology
etymology
Best Answer
Etymonline says this about that: