I listened to a song which started like "I had me a boy." Why are there two subjects, I and me? Or why did they use I and me?
There's only one subject here, I, but there are two objects. Boy is the direct object, and me is an indirect object, designating the recipient or beneficiary of the action just as it does in He gave me a ring, He bought me a ring. As @Peter suggests, this unusual ditransitive use with have suggests something along the lines of "I had a boy of my very own".
Linguists are actually still trying to understand this construction!
Stacey Conroy suggests that such "personal datives" are simply reflexive pronouns. However, Corinne Hutchison and Grant Armstrong build upon Conroy's works and argue that "personal datives" are reflexive pronouns that mean that the verb satisfies the subject in some sense.
In other words, this means something like "I had a boy, and it was satisfying to me".
I had me a boy.