MEANING: noun: A source of inspiration.
verb intr.: To be absorbed in thought.
verb tr.: To think or say something thoughtfully.
noun: A state of deep thought.

ETYMOLOGY: For the first noun: In Greek mythology, the Muses were nine goddesses, each of whom presided over an art or science. A museum is, literally speaking, a shrine to the Muses. Earliest documented use: 1390. Some other words related to the Muses are terpsichorean and calliopean.
For the rest: From Old French muser (to meditate, to idle). Earliest documented use: 1500.

MAUSE - my mother stops for a moment, then continues

MUSET - (mathematics) the one after a Lambda-set

MUSEM - how to keep li'l kids a-grinnin'