Given the following form fields:
What are the pros and cons of having the Name as a required field? (The Email and Content are both required)
In my opinion all three fields you mentioned should be required in a contact form if you are looking to reply to the message. Typically the name field is the easiest out of the three fields to fill out, therefore if you're already making the other two fields required then you might as well include the name as well.
I don't think there is a reason why users won't provide their name when trying to contact you, however they'll just make up a fake name/email if they really don't want to let you know who they are anyway.
As always it depends on the context. But typically a name is provided as a courtesy so that I know how to address you when I'm replying to your message rather than saying "Dear Mysterious User, thanks for contacting us!"
Worst case scenario is that you require it, the user wishes to remain anonymous and provides a fake name.
In reply to resorath, email is required so that you have a way of actually replying to whoever is using the contact form.
One reason I have heard for keeping the name required is to do with spam detection. If the name is missing from the email, it has a greater likelyhood to be interprested as spam.