All of the research questions listed above are potentially useful for developing a paper about Odysseus, but some are more effective and focused than others. Here's my assessment of each question:
1. "Where was Odysseus born? What are the names of his parents? Where did he travel on his journey?" - This is a set of three questions, each of which requires some basic biographical research to answer. They are useful for establishing important background information about Odysseus and could help provide context for specific events in his life, such as his travels. However, they don't necessarily lead to a clear and focused thesis for a paper.
2. "What is the name of Odysseus's biggest battle? What is his most important quality?" - These two questions are somewhat disconnected from each other, and while they could be useful in developing different aspects of a paper about Odysseus's character or achievements, they don't necessarily relate to each other in a way that would lead to a single clear thesis.
3. "What happened in Odysseus's attempt to capture Troy? What impact has this had upon modern life?" - This is a focused and potentially fruitful research question that could lead to a clear and compelling thesis. The first question requires research about a specific event in Odysseus's life, while the second question asks about the broader impact of that event. Both elements could help in developing an argument about Odysseus's significance.
4. "Who was Odysseus? Why is he important? How has he influenced today's society?" - This set of questions is very broad and open-ended, which could make it difficult to come up with a clear and focused thesis. However, it could be a useful starting point for exploring different aspects of Odysseus's character and legacy, and could lead to interesting insights about his continuing relevance.
In conclusion, while all of these research questions have the potential to be useful in developing a paper about Odysseus, the third and fourth questions are perhaps the most effective in terms of leading to a clear and compelling thesis.