Speaking as someone who has had a musical training, I think you may mean
3A musical composition arranged for performance with instruments or voices differing from those originally specified:
‘Mozart’s symphonies in arrangements for cello and piano’
New York-based AlfieProducer plays a piano arrangement of 42 popular songs like Robin Thicke’s “Blurred Lines,” Daft Punk’s “Get Lucky,” and “Radioactive” by Imagine Dragons in this impressive video.
This term is used for most cases. The main exception occurs when most of the piece is extemporized in which case it might be called an improvisation on a theme, e.g. Improvisation On A Theme (Pachelbel's Canon In D Major) performed by David Lanz https://www.youtube.com/watch?v=qqGrB4MxdsY
In the classical world you will also hear the term variations on a theme. Usually a well-known melody is played followed by many different versions of it - often virtuosic in nature, e.g. Brahms, Variations on a theme by Haydn for two pianos, op. 56b https://www.youtube.com/watch?v=PeYEcZQvBkU