Meaning of Jenna. English name. In English, the name Jenna means-small bird.The name Jenna originated as an English name. The name Jenna is most often used as a girl name or female name.English Name Meaning-small bird. Origin-England.nglish name. In English, the name Jenna means-small bird.The name Jenna originated as an English name. The name Jenna is most often used as a girl name or female name. English Name Meaning-small bird.