Why is Max known as "Mad Max" in the Mad Max Franchise?

Now first of all, this notion is likely a primary remnant of the first movie, which was of course called Mad Max. The following movies simply built upon this established character and his world and explored vastly different stories and environments. So to understand this title, the best place is to look at the first Mad Max movie alone, since the others are simply called that way because, well, they're Mad Max movies (and the "mad" still has its place when seeing how the whole world has gone mad).

So what makes him "mad" in the first part? This is much easier to answer, since in contrast to all the following movies the first part actually puts Max into a world that still worked to some degree, he had a family and a job in the police force. But during the course of the movie the circumstances and the crumbling of this world, ultimately leading up to the death of his family, slowly drive him mad. Maybe not "mad" in the sense of insane (though, to some degree maybe too), but definitely "mad" in the sense of angry. He goes onto a mad revenge quest and all that made him Max Rockatansky is either taken from him or sacrificed in favour of Mad Max.

It is ultimately a brutal revenge B-movie about a man running wild along with the world around him and for this the short and meaty title "Mad Max" hits the nail on the head. Even if Max's behaviour might not fit any strict definition of insanity, he's still going friggin' mad at the world! ;-)

Below follows my answer to a similar question over on the Science Fiction & Fantasy Stack:

In the first Mad Max movie, he is named thus by the last remaining member of the motorcycle gang that killed his wife and son, shortly before Max kills him in revenge.

As Max casually walks away, Johnny started pleading, then laughing, calling Max "...mad! Yer MAD!"
_Wikipedia

Clearly, the answer to this comes in the end of the first movie. The thing that matters most to Max is his wife and family who he nearly loves and who dearly love him. Throughout the movie, we see Max as a good cop who gets the job done, who does not deliberately set out to kill any of the criminals (Nightrider killed himself by choosing to drive past a dangerous roadblock in order to escape arrest) and whose boss admires him. We see his wife worried about him going to work and Max himself worried about becoming hooked on what he does. He quits the violent world of an MFP cop and spends time with his family.

But the bikers catch up and kill his family, leaving Max depressed and angry (i.e. mad = furious). He sets out to avenge the deaths of the people who meant everything to him by going out and killing each of the bikers one by one. His boss did not even attempt to stop him clearly because he too was happy Max was back and likely to give the MFP exactly what it wanted: an end to the biker gang once and for all!

In subsequent films, Max is depressed sometimes and does not have interest in being anywhere only the wasteland. But he always helps others. Interestingly, Furiosa (Charlize Theron's character) is also a play on a word for angry (fury) and she too had a similar backstory to Max.

I would say why Max is called Mad Max is because he chooses to live in the wasteland and not in the civilized cities is because he lost his family. This point is set in the second film when he chooses to drive the tanker as the leader confronts him telling him he's a maggot living in the real world. Looking at Max's face it's true but it's the life he wants to live. Saying that it is a title he makes. Also since it's a foreign film the title of the film may have been changed to those for American viewers.