"The high-profile singer-songwriter, musician and musical theatre actor brings significant experience as an artist to the board," he said in a statement on Friday.
The recently-retired managing director of Channel 7 Perth, Mario D'Orazio, has also been appointed to the board, with deputy chair Lee-Ann Tjunypa Buckskin set to retain her post.
Board members are paid $32,880 each year, while the deputy chair will earn $44,840.
The prime minister has not been shy in recent years about his fondness for Arena's music.
"My interest is unhealthy, bordering on unhealthy, but she's great," he told a local radio station in Perth in October.
Opposition Leader Bill Shorten said he also like her music - but perhaps not to the extent of his rival.
"I'm not the sort of fanboy that the current prime minister is, but that's fair enough," he told reporters in Melbourne.
"She's a very credentialed Australian."
Mr Shorten said in general, such appointments should be as transparent and completed at arms-length.