A minority group of adherents showing extreme support for a political movement or a set of beliefs.
Theodore Roosevelt’s presidential address to the American Historical Association in December 1913, included extracts from his essay History as Literature. That is the point that he is widely credited with coining the term 'lunatic fringe':
There is apt to be a lunatic fringe among the votaries of any forward movement.
On the face of it there's little more to be said and 'lunatic fringe' need not concern us here - it isn't even an idiom in the technical sense, in that it can be understood literally just by knowing what the words 'lunatic' and 'fringe' mean. However, there is a little more to add.
The first thing to be said is that the expression came into being with an entirely different meaning to 'extreme minority group' and that Roosevelt played no part in coining it.
In the USA the styling of hair so that it lies square across the forehead is called bangs. This is a modified form of 'bangtail' which is the name for the cutting of a cow's tail to form a straight-ended tassel. In the UK, where I live, we don't habe bangs, we have fringes. Of course, 'fringe' has been in the language for centuries with the meaning 'ornamental border' - it is known in print since the 1400s. It was co-opted into hairdressing parlance quite late on. An early example of it comes from The Alnwick Mercury, January 1873:
For pushing young maidens of twenty-five or forty a cunning fringe of hair on the forehead, in poodle-dog style, is the proper thing.
Alnwick is in Northumberland, England but, as the above piece was headed "Cuttings from American paper", we may infer that the fashion started in the USA.
Cutting one's hair to resemble a poodle wasn't universally welcomed and items like these soon began appearing in US newspapers:
Oliver’s Optics Magazine, February 1874:
“The girls!” exclaimed Miss Lizzie, lifting her eyebrows till they met the “lunatic fringe” of hair which straggled uncurled down her forehead.
The Wheeling Daily Register, July 1875:
“LUNATIC Fringe” is the name given to the fashion of cropping the hair and letting the ends hang down over the forehead.
The hairstyling terms 'fringe', 'bangs' and 'lunatic fringe' all began to be found in print around the 1870s, so it isn't at all clear how they relate to one another. The essential point though is that 'lunatic fringe' was a hairstyle.
Why might a fringe have been labelled 'lunatic'? It could be that people thought the hairstyle to be outlandish and wanted to denigrate it. This wouldn't be surprising, each generation of youth has it own hairstyle as it has its own music - to distinguish itself from the ancient creaky fogies who are twenty years older. Another theory is that it was associated with lunatics and was thought to resemble hairstyles they adopted. Having done some research on images of asylum inmates in the 19th century I can't find anything to support that view. Inmates of asylums didn't sport fringes, lunatic or otherwise. The typical hairstyle was a centre parting. Indeed, the examples I've found show less evidence of a lunatic fringe than Teddy Roosevelt did.
Having come back to Roosevelt we can acknowledge that he gave the expression 'lunatic fringe' its popular 'political movement' meaning, even if it was coined forty years earlier by an unknown hairdresser.