Monologue writing is one of my favourite exercises. Unfortunately, although they always get a positive response from readers/listeners, publishers tend not to be interested I've found - unless your name is Alan Bennett. But that's not a valid reason for not writing them.
If you're writing a novel, one way to get inside your character and find out what he's really about is to write a monologue for him. Putting your character into a situation and letting him/her work it out is a great way of getting to know their flaws and gifts and while the incident may not be specifically relevant to your story it will allow and help you to develop more fully the character.
For example, how does your character react when placed in a room full of strangers? What is going through her head? How does she behave? If there's someone she wants to make a good impression upon, how will she go about this? And remember you're writing from her viewpoint and her narrative may not be entirely factual. It will be true from her perspective but may not be accurate. But this is her view so it doesn't matter that it's not true but is useful as an insight into the way she views the world and her own place in it.
An alternative subject for a monologue is a well-known character from history or even a modern celebrity. The bible is also a rich source of inspiration. I like to find a lesser-known character, maybe one who only crops up once and may be nameless, and fill in some gaps. How the gaps are filled is up to the imagination. I find a traditional old Jewish woman suddenly sounds strangely like a Welsh mam ... But that's fine because this is my take on the story.
I've been fascinated by reactions to monologues I've written and read. One story I thought was poignant was greeted with laughter - it was good laughter by the way and I played up to it in future readings; more recently I read a monologue for what I thought was a sympathetic character: at the end one listener said, 'What a cow!'
So maybe writing a monologue also gives us an insight into our own characters and understanding ...