This persuasive writing lesson from ReadWriteThink uses the Beverly Cleary book Emily's Runaway Imagination as the springboard for kids to write letters to a librarian urging the addition of certain titles to the library. A Persuasion Map Planning Sheet(28K PDF)* guides students through steps similar to what is described above.
This resource shows the lifecycle of writing a persuasive letter to a child's parents about where to vacation for the summer. The PDF begins with the brainstorming, moves through drafting, editing, and publishing of the final letter.
From Writing Fix, here's a speech writing lesson that uses the mentor text Otto Runs for President in conjunction with the RAFT strategy. In this lesson, students assume to the role of a talking fruit or vegetable. Pretending that there's a "Fruit/Vegetable of the Year" election, the students will create a campaign speech that explains why their fruit/veggie is the best candidate for the job.
Phew. That was a mouthful! I didn't think 10 writing prompts would be that long...
One thing you'll notice is that all of the writing prompts provide the students with an authentic situation. These are all plausible things that could happen to a third grade student. As Paulo Freire would suggest, these are authentic. He or she will have a frame of reference to think about the question, and it's framed in terms of a person or group that the student already interacts with - parents, teachers, principals, etc.
All students would benefit from authentic and meaningful writing tasks, but this especially true of younger students who have trouble grappling with abstract concepts. If you need more ideas, you could try looking through this list of persuasive essay topics about school rules and see if you could adapt them for a younger audience.
You'll also want to think about how your students can organize their writing. Essay mapping and outlining is an equally important skill, and as your students get older they will need to be able to organize their thoughts into multiple, discrete paragraphs. An essay map, like this online graphic organizer, may useful in helping them organize different reasons to support their argument.
Good luck, and happy writing to you and your third graders!