O’Connor made a name for herself in the 1980 film Breaking Glass as a young singer-songwriter.
Artists including Pink, Lady Gaga and Kylie Minogue have credited her as influencing their careers.
Visitors to the exhibition will be able to watch the Breaking Glass film on a big screen and hear performances from saxophonist Clare Hirst and pianist Sarah Fisher, organisers said.
Both spent time touring with O’Connor and Hirst, who first met her in the early 1980s, said they had “so much fun” together.
She said they became friends after meeting at a cocktail event and described O’Connor as a “very determined person”.
“If she wants to do something she will,” she added.
Hirst said Being Hazel was a “lovely exhibition” and it meant a lot to see it happen.
“She’s such an amazing songwriter that probably people haven’t even heard half of her work,” she said.
“We were hard pushed to decide which ones to do on Saturday.”