The name Bar-Jesus in the Bible
Bar-Jesus is a "certain magician, a Jewish false prophet" (Acts 13:6), who resides with the proconsul Sergius Paulus at Paphos on the island of Cyprus. The proconsul asks for Paul and Barnabas to come over and explain things to him, but Bar-Jesus works against Paul and Barnabas. That leads Paul to rebuke the magician, who in turn becomes blind.
Etymology of the name Bar-Jesus
The name Bar-Jesus is a bit of a challenge, because very old sources (Jerome and Origen, for instance) don't think that this was really his name. How it came to be Bar-Jesus nobody knows, and neither is it clear what the man's name actually was. It all becomes more complicated when the author of Acts explains that the magician was also known by his translated name Elymas, and sure enough, nobody knows what that means either, or even from which language the name Elymas stems. It's not even clear whether Elymas translates Bar-Jesus or the word magician.
The second part of the name Bar-Jesus is obviously the same as the very common name Jesus, which is a Greek transliteration of the Hebrew name Joshua, and that name comes from the root group ישע (yasha'), meaning to save, and שוע (shawa), meaning to cry for help:
The whole name Bar-Jesus would mean Son Of Jesus, or Son Of Joshua, or Son Of Yah Who Saves.
As said, older sources don't read Bar-Jesus but Barieu or Bar-jeus, which is then proposed to correspond to the Hebrew verb יעץ (ya'as), meaning to advise or counsel. The name Bar-jeus would mean Son Of Counseling, which would neatly correlate to this man's alternative name Elymas, which might mean Wise or Sage. But then again, it might not.