A monologue is a speech delivered by one character entirely, without interruptions or response from the person listening. Usually, a monologue is used to express deep inner feelings. It's a term used to refer to drama; The Member of the Wedding was originally written as a novel, but, fittingly, it was also turned into a play, so, if you have been asked to write a monologue in Frankie's voice, you can refer to that play for some inspiration in terms of style, phrasing, and so on. You can find the whole text of the play online—please see my link below.
You haven't specified in your question when the monologue should take place. We're going to write about Frankie's real feelings, but you need to decide which version of Frankie you're going to think about. Will it be the Frankie of the beginning of the story, before the issue of the wedding even arises? At this point, Frankie is feeling extremely lonely and despondent. Her best friend has left, she is too old to sleep alongside her father anymore, she hates her hair and her gangliness, she has no other friends at school, and she feels generally out of place and lonely. So, what would this Frankie confess to her father? You might imagine her sitting him down to explain all of the above: why she is so unhappy and that she feels as if she doesn't belong and will never belong. Perhaps she might appeal to her father to console her—she is entering her adolescence, and she feels unanchored and unbalanced.
Or, you might write a monologue from the perspective of Frankie at the end of the story. This Frankie could have a lot more to say. She could express to her father all the things she felt at the beginning of the story—as detailed above—but could also tell him how she felt when the prospect of her brother's wedding arose. Over the course of the summer, Frankie felt important, because she was going to finally be a part of something everyone was interested in, and as if she were growing up—F. Jasmine Addams, no longer a child, might have a better chance of belonging to something than Frankie ever did.
Then, of course, what transpires with the soldier makes Frankie feel as if she has been burned, punished for reaching out. She is frightened by his behavior; what would she say to her father about this? Would she ask for consolation on this front, too?
And how does Frankie feel at the very end of the story? Why has she changed her name—is it a desire to fit in as another person, the need to create a new persona and reinvent herself? Does she feel more connected to the world now she has a new friend, or is she more lonely than ever without Berenice and John Henry, the only people who stuck by her the whole time?
There's a lot to think about here. I think you could easily make arguments in either direction in Frankie's voice.