Morning Dave and welcome
In short yes you can make your own mandrel, but if you already have a metal lathe you may already have the tools in your workshop to turn a pen.
Between centres has been mentioned above already, that is an excelent way to start.
Why don't you put a quick intro of your self, your set up and what you have done in our "introduce your self" part of the forum. That way we get a better understanding and can offer advice in an informed way so to help you and your son to make the pens.
By the way, if you have the lathe then you are 100% there toward turning your first pen..
By the way, Mandrels rods can be brought for less than £20