The colloquial expression mad money denotes:
– money for use in an emergency or in any unexpected eventuality;
– money that is surplus to one’s normal requirements and which may be spent on a whim.

The text containing the earliest occurrences of mad money that I have found indicates that this expression:
– originated in the slang of the flappers, i.e., of the young women who, in the 1920s, were intent on enjoying themselves and flouted conventional standards of behaviour (cf. “the cat’s whiskers”, and all that jazz);
– denoted the sum of money that a flapper carried as a precaution so as not to be left financially helpless in case she and her boyfriend got angry (i.e., ‘mad’) at each other while on a date.

The text in question, by Carl Victor Little, United-Press Staff Correspondent, was published on Thursday 2^nd March 1922 in several U.S. and Canadian newspapers—for example in The News-Sentinel (Fort Wayne, Indiana, USA):

CHICAGO, March 2.—Flapperanto—the dialect of the modern girl—has made English a dead language.
English-speaking intruders on the campus of Northwestern or Chicago universities probably would have this same experience:
“Ooh!” flapperantoed the co-ed, “I have lost my mad money.”
“Lost w-h-a-t?”
“My mad money. I had it squirrelled in my locket!”
“Meaning which?”
After resorting to every mode of expression from sanscrit to the sign language, it was discovered the 1922 girl always ‘squirrels,’ or hides, a few dollar bills known as ‘mad money.’
Thus, the independent maid need not walk home in case she becomes angry with her escort at a dance. She just takes her mad money, calls a taxi and leaves Apollo flat on the wax.
Noah Webster, if he visited flapperland, could get enough words—synonyms and antonyms included—to compile a $25 dictionary.
Abridged edition of flapperanto follows:
Button shining—close dancing or achieving the same effect without the music.
Mugging match—a petting engagement; to spark; to spoon.
Necking party—see mugging match.
Pash stuff—emotional torridity.
Jewelers—flappers who measure college success by the number of fraternity pins they collect.
Monogs—taken from the old English ‘monogamist,’ referring to the male or female student who plays with but one person of the opposite sex.
Seraph—girl who likes to be kissed—but not violently.
Owl—flapper who cuts classes and is only seen at night at dances and parties; usually wise enough to get high grades in academic work.
Swift’s premium—clumsy flapper; wall flower; a ham.
Feature—to see; ex.—“I can’t feature him for the darkness.”—Anon.
Punching the bag—act of a man who chats with a girl—and keeps on chatting; gymnasium term perhaps referring to the social finesse of a dumb-bell.
Holiholy—flapper who won’t indulge in mugging match; obsolescent.
Holaholy—male of a holiholy; obsolete.
Dudd—profound student of books—not flappers.
Ground gripper—female form of dudd.
Baby grand—Corn-fed co-ed.
Pocket twister—girl who eats, dances and drinks up all of a man’s spare change.
Struggle—a dance.
Pill—professor.
G. G.—Refers to a man; coded form of the English expression, gullible goof, which speaks for itself—but he doesn’t.

In Slang To-Day and Yesterday (London: Routledge & Kegan Paul Ltd, 1933), the New Zealand-born British lexicographer Eric Honeywood Partridge (1894-1979) wrote that, during the First World War, New Zealand soldiers used the expression mad money of the return fare that an English girl carried in case her soldier friend became too free in manners (i.e., ‘mad’):

Some idea of New Zealand slang can be obtained from the following short list of words employed by those who served in the War and by many persons still: New Zealand slang is perhaps the most conservative of all the colonial slangs: […] mad money, return fare, it being very generally believed by the New Zealand troops (Fernleaves or, as they preferred to call themselves, Diggers) that every English girl infallibly carried her return fare in case her soldier friend became mad, i.e., acted with an excessive freedom of manner.

However, in A Dictionary of Slang and Unconventional English (London: Routledge & Kegan Paul Ltd, 1951), Eric Honeywood Partridge added reservations to his original statement:

mad money. A girl’s return fare, carried lest her soldier friend got ‘mad’, i.e. too amorous for her: New Zealand soldiers: 1916–18. Mostly a legend, and concerning only English girls.