If I were leading this exercise, I would go with the second approach, use the ideas presented to generate more ideas. Once all the ideas are gathered then discuss and work on improving ones that the group feels are the strongest.

Is there a way that the playing field can be leveled so ideas are presented without being attached to a person? I've seen great ideas passed over because the person running the session doesn't like the person who proposed the idea. It's not just extroverts dominating the conversation, it can be political or social.

I agree. Bad ideas can beget good ones. For example, I can be impatient. I know it's a bad idea to eat raw bacon but in the morning I don't want to cook bacon because it's messy and time consuming. How do I get around this? Create pre-cooked packaged bacon. A very simple example but there's now several choices of pre-packaged, pre-cooked bacon at the grocery store.

These are always excellent questions to ask when ideating. Who am I targeting? What is my idea? When will my target use my idea? Where will they use it? Why will they want to use it? How will they use it. If those question's can't be answered, then the idea needs to be re-scoped to ensure that each of those parameters can be clearly defined.

"What ifs" are some of my favorite ways of brainstorming along with, "yes and." What if this is a bad idea? Yes, and what if it's not a bad idea, just an idea that needs a yes, and?

I agree with iterating and refining. As move things through the design process, whether it's a project for work or school or one of my personal knitting projects, I enjoy going back over the project to see where I made mistakes, what went well, and where I have room for improvement. Many times where I end up is not at all where I started.