late 15c., "to keep up, maintain, to keep (someone) in a certain frame of mind," from Middle French entretenir, from Old French entretenir "hold together, stick together, support" (12c.), from entre- "among" (from Latin inter; see inter-) + tenir "to hold" (from Latin tenere, from PIE root *ten- "to stretch").
Sense of "have a guest" is late 15c.; that of "gratify, amuse" is 1620s. Meaning "to allow (something) to consideration, take into the mind" (of opinions, notions, etc.) is 1610s. Related: Entertained; entertaining.