What is the term for someone who has a last name that can also be a first name? For example, Brian Stella.
I've always heard this called a "reversible name" - but that's referring to the forename and surname together.
Here is a list of them. http://richard.tangle-wood.co.uk/reverse.html
I'm pretty sure there is no specific term to describe the generic concept you're describing. The closest specific term is probably a pseudo-surname where a child takes on a parent's first name as their last name. This is referred to as a Patronymic or a Matronymic name.
Patronym - a family name derived from name of your father or a paternal ancestor (especially with an affix (such as -son in English or O'- in Irish) added to the name of your father or a paternal ancestor)
During the American Civil War, soldiers began to use the surnames of their commanding officers (Grant, Lee, Sherman) as first names for their sons. This, and the American custom of using the mother's maiden name as a child's given name, along with the general twentieth century trend towards 'new' names, has led to many other surnames being recognised as first names. Thus Old English and Celtic personal names, long out of use except in their adaptation as surnames, have returned to the central stock. This was formerly almost exclusively confined to boys' names but is now beginning to produce many new girls' names, especially in America. The more established surname adaptations are subject to the same spelling variations as other first names. The link
I guess you could say they're interchangeable surnames and forenames. Here's a link with many examples: http://www.nationalfinder.com/surnames/index.htm#INTERCHANGE
Or you could also use interchangeable surnames and given names.
And apparently, on further Internet searching, there is the term "firsty firsty" for this. See this blog as an example: http://firstyfirsty.blogspot.com/2008/02/curse-of-firsty-firsty.html
I thought firsty firsty might just have been a term used by kids, but this appears to be evidence of wider use in popular culture.
Note that there is a corresponding lasty lasty.