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.
I don't think it matters so much where you mention your potential advisor, as long as you do it very, very prominently. From what I understand the fact that you already have an advisor in mind is a key fact in the application procedure, and you should never assume that the evaluation committee reads your letter with care. That is, format / write it in a way that even an evaluator who only glances over your letter will understand this fact.
With that in mind, I suggest to bring up your advisor (at least) in your cover letter, in the introduction, and maybe in the conclusions of your research statement. You don't need to be heavy-handed about this - it is ok if you add something along the lines of "I am very much looking forward to work with Prof. X on Y" in a few places, although at least once in your statement you probably want to have a small dedicated paragraph where you discuss that you have been encouraged by Prof. X to apply to work with them.