Diana is neither Roman nor a goddess. She'a an Amazon of presumably Greek descent. Although her powers are gifts from the Olympian gods (definetly in the comics, presumably on the show), she herself is mortal. She was for a time elevated to godhood in her comic book, but that didn't last long.
Originally Posted by Enchantress
The goddess Artemis is one of her patron goddesses, although it is not known (at least on the show) where the name Diana came from, since that would be of Roman origin and thus foreign to the Amazons.
In the comics, she is named after Diana Trevor (Steve's mom) who ended up on Paradise Island, and died a hero.
"I didn't think it was physically possible, but this both sucks and blows." B. Simpson