The BIG one is: What, if anything, does the name Bananarama even mean, and how did they come up with it? The only context where they are even seen with bananas is in the "Cruel Summer" video.
I know for a fact that a lot of the name was inspired by Roxy Music's song "Pyjamarama", because they were all big Roxy Music fans. I believe I read somewhere that they got the "banana" bit from the '70s cartoon "The Banana Splits" because they had all been fans of the cartoon as children, but that's on shakier ground than the Roxy Music influenced bit.
The name was also inspired by the 1964 song The Name Game by Shirley Ellis. "Shirley! Shirley, Shirley bo Birley Bonana fanna fo Firley Fee fy mo Mirley, Shirley!"