This kind of piece is really a theme and variations. Another common type of piece is the Prelude and Fugue, two movements. The first is free-form, often melodic like a short sonata, or constructed of chords; the second is made like a round, like Row, Row, Row Your Boat except WAY more complicated.
Borne [borne] as a boys' name has its root in Middle English, and Borne means stream. Borne is an alternate form of Bourne (Middle English): place name.
Borne is now the standard past participle of bear in all situations except the expression to be born, meaning (as you say) to be brought into the world. See earlier in the thread, and in particular panj's quote from the OED.