Go for a practical, close-to-home character modeled on yourself, someone in your family, a friend or acquaintance. When you build a character's monologue from someone you know personally, you will access that person's speech pattern easily and naturally. Write about a guy on a park bench watching the autumn leaves drift to the ground in Central Park. Or strand your character on a raft in the middle of the Pacific Ocean. Make sure to be specific about where your character is and what the reason is that he is giving the monologue. For example, the guy stranded on a raft could be screaming at a distant, passing ship for help or coaxing a circling shark to beat it. Use an everyday, conversational style of language to convey the realistic style of speech of these characters.
Alternately, go for an outlandish character such as an alien, a ghost or a disembodied voice from a galaxy and time far in the future. For these characters, you'll need to employ a great deal of imagination and/or research to develop the way the character thinks and speaks. Using blank verse or a stilted speech pattern will help accent the character's otherworldliness. You might write about a wise man or woman from the fabled city of Atlantis who has mysteriously appeared at a modern-day college campus. Or the same college campus could be the site of the appearance of the ghost of a former student or teacher. If you're using an exotic character, consider placing them in a commonplace setting, so that you can explore the juxtaposition between the mundane circumstance and the exoticism of your character in the monologue.
Historical characters offer a wealth of material on which to base a monologue. One example would be a monologue written from the perspective of a Puritan woman accused of being a witch in sixteenth-century New England. Heighten the drama of the moment by locating her monologue at the moment of her condemnation, pleading for her life to her judges. Or have her monologue be spoken to the jailer that leads her to her death, possibly casting a spell upon him so that he might release her. Other historical characters to write about might include Helen Keller at the moment she understands sign language or Marie Antoinette as she awaits her fate in prison. Researching speeches and diaries written by the person or during the time period you have chosen can help create an accurate sense of voice.
You can also use animals, plants or inanimate objects as the subject of your monologue. Write about a seagull who is sitting on the shore as the spilled oil drifts onto the Louisiana shoreline or write from the point of view of an endangered species such as the tigers of Asia; write from the point of view of a much-loved and now abandoned toy or create a monologue for an ancient but dying tree that is just about to be cut down. Researching the style of speaking used by others --- such as how the toys speak in "Toy Story" or what sort of voices are given to the trees in "Lord of the Rings" --- provides concrete examples of the dialogue of inanimate objects to use as inspiration.