The phrase that’s the way the mop flops means that’s the way the situation is, and it must be accepted, however undesirable—synonym: that’s the way the cookie crumbles.

Partially based on the identical sounds in mop and flop, this phrase seems to have originated in U.S. teenagers’ slang—cf. also see you later, alligator.

The earliest occurrence that I have found is from the column Gone Gab Corner, in the teen scene section of the Democrat and Chronicle (Rochester, New York) of Monday 5^th September 1955:

Todays [sic] guest in the Gone Gab Corner was a staffer in the trail department at Camp Pioneer, Boy Scout Massawepie Camps, this summer.
That’s the way the mop flops!—“A fatalistic philosophy of life.”
I’ve been out on a snipe hunt—“I was sent out on a wild goose chase.”
A gopher—“That’s a waiter, here. He’s called that because he has to go-fer this and go-fer that.”
Life is a dangerous game—you never get out of it alive—“What you say when something goes wrong.”
That went over like a lead balloon—“When a joke falls through.”
Glenn Ireland Troop 58

The second-earliest occurrence that I have found is from More Military Jobs Open to Service Girls, by Ruth Cowan, published in the Okemah Daily Leader (Okemah, Oklahoma) of Wednesday 9^th November 1955, about a “pamphlet, “Careers for Women in the Armed Forces,” jointly released by the departments of Defense and Labor”:

A woman gets the same pay and benefits as a man on the same job. She may get to travel abroad. She may get officer status. But all is not gravy.
The pamphlet cautions: “She will work hard. She must go where she is sent and do what she is told, for those in military service are under orders.” In other words, you have take it whichever way the ball bounces (or the mop flops).

Vivian Brown recorded the phrase in her column Young Moderns, published in The Muncie Star (Muncie, Indiana) of Sunday 19^th February 1956, and in many other U.S. newspapers the following days:

“That’s the way the mop flops” is added to expressions of Eastern young people. It’s supposed to take the place of such antiquated phrases as the “way the wind blows” etc.

On Saturday 2^nd November 1957, the Tampa Morning Tribune (Tampa, Florida) published an interesting article about Hillsborough Council high-school pupils’ slang:

Teen Talk—
Ungawa! Only ‘Freets’ Don’t Dig Latest Expressions In Teen Land
By Ellen Edmiston
Tribune Correspondent
[…]
Plant students enjoy two very popular teen-age entertainments, talking and eating. “Ohy gang” is a greeting used in place of such older phrases as “Good morning” and “Hey!” “I care” is another way for saying “I don’t care.” “What” is a word used whenever a person feels the urge to utter it. “Oh well must” is an abbreviation for “Oh well must be nice,” “Gomn” saves the time of saying “Gets on my nerves,” “Oh you’re right” has the opposite meaning and “Get it away” is self explanatory. Older phraseology still in use includes “Great Scott” and “Swell.” A “Complete Scubo” stands for “Such a cute boy, oh!”
[…]
Expressions of excitement at Sacred Heart are “Caramba!” “Good grief” and “I tell you.”
[…]
[At] Academy of Holy Names […] slang phrases include “Don’t panic,” “fantabulous,” “neat as cob” and “how revolting.”
[…]
“Ungawa” is an expression of excitement at OLPH [= Our Lady of Perpetual Help], “a blast” is “worse than an insult,” “a freet” is “worse than a creep.”
Large charge to Turkey Creek pupils is “So what?” “Big deal, Lucille” is another way for saying the same thing. “Touch you” calls down people who are bragging, and “Cornier than corn” is descriptively clear.
[…]
“Aloha” is “hello” at Jefferson. “Sharp as heck,” “oh man” and “shoulda” and “hafta” are heard all over school. […]
Interpretations at Brewster: “Slip the culture to me vulture” means “explain it.” “That’s the way the mop flops” and “See what I mean, Jelly Bean” put color in the conversation.
At East Bay High, “Goofin’ around,” “just a goof off,” “odd pod” and “bird egg” all describe “a goof.”

This photograph illustrated Ellen Edmiston’s article in the Tampa Morning Tribune of 2^nd November 1957:

Slang Reigns—Pert Paula Gunn of Brewster School tells a classmate, “That’s the way the mop flops,” in explaining the popular use of slang expressions among teen-agers. However, Kelly Nichols, a complete Scubo (such a cute boy, oh), would turn into an ‘odd pod’ if he took slang literally.