Use of a Mythical character's name

A customer is considering using the word "Sauron" (The mythical all-seeing eye from Lord of the Rings) as PART of a company name, it fits extremely well with the purpose of the business.

I have been asked to start designing a logo and website, the use would include both text and a logo incorporating an eye.

I have seen this word used before, and is currently used in other contexts, including in other fictional works.

Is there any reason I cant do this? Would I be infringing any copyright / trademark / intellectual property rights?