Mario's creator, Shigeru Miyamoto answered this in an interview with GameInformer a while back.

To him, Mario and Luigi don't have last names, but in the film version, Mario and Luigi have the last name of "Mario".

“This is an old story, but Hollywood did a film version of the Mario Bros. many years back. There was a scene in the script where they needed a last name for the characters. Somebody suggested that, because they were the Mario Bros., their last name should be Mario. So, they made him “Mario Mario.” I heard this and laughed rather loudly. Of course, this was ultimately included in the film. Based on the film, that’s [how] their names ended up. But, just like Mickey Mouse doesn’t really have a last name, Mario is really just Mario and Luigi is really just Luigi.”

In 2015, Miyamoto reportedly confirmed that the brothers' last name is officially "Mario", but I have been unable to find a reliable translated source. Most news sources cite this article about the press conference; this is the translation courtesy of ixrec.

任天堂公式の名称は「マリオ・マリオ」と判明！ The official Nintendo name for him has been confirmed to be "Mario Mario"! 宮本氏はステージにあがり、「マリオ・マリオが本名」と発言したのである！ Mr Miyamoto came up on stage and stated that "Mario Mario is his real name"!

(note this might be a paraphrase rather than Miyamoto's exact words)