Where should you mention the name of potential mentor in research statement when applying for a post doc? I am going to apply for a postdoc position. I exchanged emails with one of the  professors as potential mentor and he told me to mention his name in my research statement. In which part of the research statement his name must be written? Is it enough to write it in the first paragraph?
Your research statement should probably mention the new potential mentor's name in the context in which his research interests align with yours. So give a bit more than just the statement that he has agreed to mentor you. In other words, say why it would be a good plan for you to work with him. 

As to having your previous advisor give you advice on your research statement, there is no ethical concern at all. It is natural for you to continue to seek professional advice from your doctoral advisor. Your relationship doesn't break when you finish your degree. It changes a bit, as you become his colleague in the field, rather than his student.