There is probably more than one factor involved here, but surely increased stress plays a crucial role. It is difficult to say our name because we know, consciously or subconsciously, that we can't word-substitute our name, and this lack of choice increases stress tenfold.
There have been PWS who have changed their name because of the difficulty of saying it - and then later started to stutter on their new name; once again because they HAD to say it. They did not have space, freedom and alternative options for saying their name. That's how stress works - a lack of options is an important source of stress.
Fact is, saying your name remains difficult, not only because of the lack of options, but also because you have stuttered countless times on it. If you stutter lots of times on a particular word or sound, a "stutter pathway" is created in your brain. This happens by means of the process known by psychologists as "conditioning" or "learning". When you want to say that particular word, like your name, a "reminder warning" pops out from your brain, "warning" you that you stuttered on that word in the past. This warning will tend to stress you so that you actually stutter on that word.
A possible solutionIf you have learned to use a fluency technique, there may be a solution to the name problem. Dr Martin F Schwartz recommends the following procedure, which worked very well for me. I quote from my book Coping with Stuttering:
"If you have problems with saying your name (or any other particular sound or word(s) which you need to say frequently and makes your life a misery, such as your home address, telephone number, a standard work-related telephone greeting etc.), you should devote four focused days to eliminate this problem once and for all. Use the Passive Airflow Technique (or any other effective technique) to say your name as much as possible every day for four days. Vary the exercise with full sentences, e.g. ‘My name is (name)’. Approximately four thousand applications – one thousand per day for four days – should remove much or all of the word stress associated with your name. Of course, you don't actually have to count the number of applications. Just fill those days with repeated, continuous applications.
"This procedure is based on the reconditioning principle in psychology. By saying your name fluently so many times, you 'overwrite' your previous 'programming' lodged in your brain. Because of your stuttering in the past, your brain contains 'stutter pathways'. But when you say your name fluently so many times, you overwhelm and overwrite those stutter pathways and replace them with fluent pathways. Psychologists call it 'extinguishing' the old conditioning.
"I personally found this procedure to work very well. I actually took some days off from work just to practise saying my name for hours at a time. It was one of the best investments I ever made in my life. One’s name is a symbol of yourself; if you can’t say it, it really damages your self-esteem, so it pays to work on this. I know it's a boring procedure, and demanding - but the effort should be well worth it. Actually it's a good sign that you get bored, because that shows that the reconditioning is taking effect. The old fears are being replaced by boredom. At the end of my four days of doing this I was completely sick and tired of saying my name - and I know that I will never again stutter on it because I created a fluent pathway to it."